So far as I can make out, the EU's treatment of the Channel Islands is not effectively hindered by the UK's membership of the EU.
If so, Brexit will not gain the EU much leverage that it doesn't already wield.
the Channel Islands are more or less the personal property of the British Royal House
They are not personal property.
The Channel Islands are "crown possessions" but the crown is a metonym for both the state and monarch.
The concept of the Crown developed first in England as a separation of the literal crown and property of the nation state from the person and personal property of the monarch.
are member of EU's common market by proxy of UK's EU membership
That's not quite right, and the details matter.
the Islands are part of the Customs Union and are essentially within the Single Market for the purposes of trade in goods, but are third countries (ie outside the EU) in all other respects.
EU and the Channel Islands
Jersey has a special relationship with the EU through the UK. We are only regarded as being a part of the European Union for trade in goods, otherwise the Island is not a part of the EU. The formal relationship is set out in Protocol 3 of the UK's 1972 Accession Treaty, and you may hear this called the ‘Protocol 3 relationship’.
Brexit and Jersey.
Note "trade in goods" - I interpret that as excluding trade in services (such as financial services)
The EU apparently treats the Channel Islands in the same way asfor non-EU states when attempting to restrain the actions of tax havens.
Of the jurisdictions with links to the UK – Bermuda and the Cayman Islands, along with Guernsey, Jersey and the Isle of Man – have been placed on a so-called “grey list” who have committed to reform their tax structures to ensure, for example, that firms are not simply using their 0% corporate tax rates to shield their profits.
It is understood the British government tried and failed to ensure those jurisdictions would not be screened by the EU’s tax experts but was overruled. A further eight jurisdictions affected by recent hurricanes will be addressed in February.
So the UK's current status as an EU member is no shield for the Channel Islands in that respect.
the Channel Islands will not enjoy the EU's basic freedoms, especially free movement of capital
Do they enjoy them?
5. Relationship to the European Union
The Islands have a special relationship with the European Union provided under Protocol 3 to
the UK's Treaty of Accession to the European Community. Under Protocol 3, the Islands are
part of the customs territory of the Union and therefore Union customs matters, the common
customs tariff, levies, quantitative restrictions and any measures having equivalent effect apply.
There is free movement of agricultural goods and derived products between the Islands and
the Union. Also included are measures relating to the trade in agricultural goods and derived
products with third countries.
However, other EU Rules do not apply to the Crown Dependencies. Implementation of the
provisions on the free movement of persons, services and capital is therefore not required, and
the Islands are not eligible for assistance from the structural funds or under the support
measures for agricultural markets. EU tax instruments do not apply, nor do the developing
justice and home affairs initiatives or the Schengen acquis, although the Islands support
improved judicial co-operation within Europe and have also voluntarily applied for recognised
equivalent status in a number of key law and policy areas.
Fact sheet on the UK’s relationship with the Crown Dependencies