Being a minority government is immaterial Pops!
The PM can write a letter to the EU , worded in such a way as to request an extension and thereby go along with ‘the law’ as you put it, but which he will make absolutely clear can be revoked if ,no , let’s be positive, not if, ‘when ‘ he wins a Majority at a General Election.
Can you really see the EU replying with a yes, because they know that there is every chance of him winning an election, and them thereby shooting themselves in the foot, all 27 feet?
His platform will clearly be (either before or after 31 October) that he still stands for taking us out of the EU, but Labour, the LiDems and the SNP prevented it , and frankly I think that playing the ‘blame game’ will get him a substantial majority
The EU are just starting to realise that they will get no deal if they don’t back down- they have already agreed that this will cost individual states huge sums and job losses etc etc, with massive bailouts forecast- it has been a mistake all along to promote that the UK will be the only loser in a no deal.
So, it is likely that BoJo will get a deal ( of sorts ,admittedly ) which will allow trade talks etc to begin.