Do you mean, you can live in France and pay tax on your UK state pension in the UK and expect France to let you off paying more tax in France?
Well no, because it is tax that is due to France.
France would claim from you the tax it is entitled to, and it would be up to you to reclaim the tax that you wrongly paid in the UK. There is no mechanism for France claiming an individual’s tax back direct from the UK.
Tax has to be paid to the right tax authority. You may not feel it makes much odds to you where you pay tax, but it makes a lot of difference to each tax authority whether they receive the tax money due to them or not.
If you owed £1000 to Peter, but you decided to give it to Paul instead because you like Paul better, is Peter going to say Oh well as long as you’ve paid it back to somebody that’s OK, forget about it?