Brythan covered the 2nd part (how does Parliamentary sovereignty interact with that alleged rule which limits initial parliamentary security if it exists). And it does exist, but only to some extent
Part 2 of the Constitutional Reform and Governance Act 2010 requires the Government to lay before Parliament most treaties it wishes to ratify, along with an Explanatory Memorandum. This gave statutory form to part of a previous constitutional convention on parliamentary involvement with treaties (the Ponsonby Rule).
The 2010 Act also for the first time gave parliamentary disapproval of treaties statutory effect, and effectively gave the House of Commons a new power to block ratification. The process is this:
The Government may not ratify the treaty for 21 ‘sitting days’ (ie days when both Houses were sitting) after it was laid before Parliament.
If within those 21 sitting days either House resolves that the treaty should not be ratified, by agreeing a motion on the floor of the House, the Government must lay before Parliament a statement setting out its reasons for nevertheless wanting to ratify.
If the Commons resolves against ratification – regardless of whether the Lords did or not – a further 21 sitting day period is triggered from when the Government’s statement is laid. During this period the Government cannot ratify the treaty.
If the Commons again resolves against ratification during this period, the process is repeated. This can continue indefinitely, in effect giving the Commons the power to block ratification.
Neither House has yet resolved against ratification of a treaty under these provisions, and there are limited options for how they can do so.
Despite looking like a major change, the provisions of the 2010 Act have several exclusions and limitations [Lists some types of treaties that are excluded from the 2010 Act, but this list doesn't include trade agreements.]
No requirement for debates or votes
Although the 2010 Act puts on a statutory footing Parliament’s opportunity to scrutinise treaties, it does not require Parliament to scrutinise, debate or vote on them (and it rarely does so).
There have been some calls for a process that results in more debates and votes on treaties, perhaps involving the committees, but Parliament has so far been reluctant to set up new mechanisms for treaties.
This is in contrast to many other countries where parliamentary approval is required at least for certain defined categories of treaty. Even some other ‘dualist’ countries have incorporated some kind of parliamentary scrutiny of treaties, for example Australia which has a dedicated Joint Standing Committee on Treaties.
Parliament can only oppose (or tacitly accept) a treaty in full – it cannot amend treaties.
There is no general requirement or mechanism for parliamentary scrutiny of (non-EU) treaties while the Government is negotiating them. So Parliament is not usually involved at the stage when changes could still be made to the text of a treaty.
This is fairly typical; the US is rare in allowing the Senate Committee on Foreign Relations to propose amendments to treaties.
There have been several proposals for parliamentary involvement before signature, to minimise disagreements when it comes to ratification, but there is also considerable opposition to such ideas.
However, Brexit has re-awakened the debate on how Parliament should be involved with treaties. [...]
So the Trade Justice Movement criticism is fairly correct in how it describes the facts. Of course, as Brythan notes, the Parliament can change its mind and pass a different law for how it is supposed to deal with international/trade agreements.
Also, the "dualist" issue needs expanding as it affected Brexit:
The corollary of the Government’s dominant role in making and ratifying treaties is the fact that treaties cannot change UK domestic law.
The UK is a ‘dualist’ state, which means that treaties are seen as automatically creating rights and duties only for the Government under international law. When the Government ratifies a treaty – even with Parliamentary involvement – this does not amount to legislating. For a treaty provision to become part of domestic law, the relevant legislature must explicitly incorporate it into domestic law.
The Miller judgment
This constitutional feature was central to the Supreme Court’s January 2017 judgment in the Miller case (about whether the UK Government needed the prior authority of Parliament in order to trigger the UK’s notification of withdrawal from the EU Treaties). The majority judgment set out ‘two features of the United Kingdom’s constitutional arrangements’:
… The first is that ministers generally enjoy a power freely to enter into and to terminate treaties without recourse to Parliament … The second feature is that ministers are not normally entitled to exercise any power they might otherwise have if it results in a change in UK domestic law unless statute, ie an Act of Parliament, so provides.
The ruling made it clear that the Government cannot make or withdraw from a treaty that amounts to a ‘major change to UK constitutional arrangements’ without an Act of Parliament:
We cannot accept that a major change to UK constitutional arrangements can be achieved by ministers alone; it must be effected in the only way that the UK constitution recognises, namely by Parliamentary legislation.
Applying the principle to this case, the judgment held that the UK Government could withdraw from the EU Treaties only if Parliament ‘positively created’ the power for ministers to do so. This was because the EU Treaties are a source of domestic law and domestic rights that ministers cannot alter using the prerogative alone.
Treaty provisions that are not incorporated into domestic law can have only indirect domestic legal effect at best. For example, where legislation is capable of two interpretations, one consistent with a treaty obligation and one inconsistent, then the courts will presume that Parliament intended to legislate in conformity with the treaty and not in conflict with it. [Gives some examples, but they are not from trade].
But usually, before the UK Government ratifies a treaty, it seeks to ensure that any domestic legislation needed to implement it is already in place.
Given how extensive trade treaties can be these days (e.g. covering non-tariff barriers), it's somewhat dubious that the UK government can implement them fully by itself, i.e. with no domestic legislation.
A more articulate criticism along the TJM lines (but not entirely convincing given how Commons opposed the Withdrawal Agreement), which expands on what I wrote in the previous paragraph is that
once the government has concluded a treaty which, in most cases, cannot easily be re-negotiated: the treaty is at that stage effectively “take it or leave it,” and those [government majority] MPs may well be reluctant to humiliate their government by telling it to leave it. [...]
the view that the traditional model is unsatisfactory has been gaining ground. The heart of the problem is that international treaties concerning trade are far removed from the Cobden-Chevalier treaty (the 1860 tariff reduction treaty between the UK and Second Empire France). It is no longer a matter of negotiating tariff reductions on wine and agricultural produce in return for tariff reductions on manufactures in the course of a few lunches and an audience with the Emperor. Modern trade treaties are vast documents including very large numbers of commitments on sensitive matters of supposedly domestic policy, ranging from food standards to data protection to immigration rules to public procurement. If the Crown can simply produce one of these vast treaties at the end of a negotiating process and say to parliament “here it is: take it or leave it,” in a context where neither rejection nor amendment is realistic, then effectively parliament has handed over to the Crown the power to legislate in a vast range of areas.
Criticism of the traditional model is particularly powerful in the case of the biggest trade treaty of them all, namely the hoped-for EU withdrawal agreement and any subsequent deep and comprehensive free trade agreement with the EU.
This was written in September 2018, before Commons rejected several times the Withdrawal Agreement.