Would there be any procedural way for the UK Parliament short of ousting the government to vote on the current negotiated Brexit deal within the next days?
No. Other than certain very limited exceptions, the government has the sole right to decide what legislation and non-legislative motions are put before the house.
For example, Commons Standing Order 14 begins:
(1) Save as provided in this order, government business shall have precedence at every sitting.
...and SO 27 says:
The orders of the day shall be disposed of in the order in which they stand upon the paper, the right being reserved to Her Majesty’s Ministers of arranging government business, whether orders of the day or notices of motion, in such order as they think fit.
In particular, only those items which appear on the Order Paper may be discussed. Its contents are decided by the government, and it's normally published the evening before the date in question.
In this instance, the Leader of the Opposition was able to apply for an emergency debate under SO 24, on the "Government’s management of the meaningful vote debate". However, such motions can only be used to express the opinion of the house, and place no obligation on the government to do anything.
It is possible that a private members' bill could be used to introduce legislation to, for example, stop Brexit. However, to have any chance of success, a PMB ultimately requires the government to provide some of its time in the schedule (as well as significant support from MPs in general), which is why so few PMBs ever become law.