You seem to be missing the point. It doesn't have to be the will of the people. It just needs to be said as such, and given the 'rebels' were already citing the survey I posted in interviews 2 days ago, they'll use whatever they can to make it look as if it is. When they add that to a run against the 'undemocratic' HoL, they might manage to convince enough people that they are defending democracy. They don't have a lot of other options.
So what you are saying is that people can just make up shit, or cherry pick selected info to suit their own political agenda. Well no shit Sherlock. Doesn't mean that any of this will actually gain any real traction does it.
If anything the recent debacle over the Rwanda scheme is shifting the polls more in favour of Labour than before.
I think that the Jenricks and Bravermans wanted the bill to be so obviously deeply unacceptable to the Lords that their 'intransigence' was guaranteed, and this 'will of the people' would definitely run. Sunak.is aiming for the same but was wanting a unified as possible Tory group of MPs to create the appearance of unity. His pitch on the will of the people now is in the hope that the Lords do knock it back.
But the 'people vs the Lords/Judges' (delete as appropriate) only works where the Lords/Judges are blocking something with credible majority support/mandate. That was the case for Brexit, it isn't the case here. Secondly it only works if it is an issue that is right at the top of the list of issues for most people. Again Brexit was at the time, the small boats isn't, it is way down the list of issues when voters are polled, below the economy, cost of living, NHS etc.
The polling that I've seen recently specifically on the Rwanda scheme (not some rather generic immigration stuff) seems to boil down to public opinion nearest to: 'FFS stop it now, stop wasting time and money on something that won't work and frankly get on with stuff that really matters to us'.