How does using the proficiency gained through beguiling influence work if it is later elevated to expertise through another class feature and then the beguiling influence invocation is then changed for another invocation?

Mechanically, this seems a bit overpowered, if convoluted, but you could also a rationalize that you have built on the proficiency from your patron and are now capable in your own right and now seek a different boon.

I could see where it could be argued to work, but I could also see the opposite.  It could be argued that while expertise doubles your proficiency bonus, it doesn't actually allow you to add it to the roll.  If you were going with that interpretation and also had jack of all trades, you would both double and halve your bonus before adding it.

In any case, it will almost certainly come down to DM fiat.

I have bad news:

Occasionally, your proficiency bonus might be multiplied or divided (doubled or halved, for example) before you apply it. For example, the rogue’s Expertise feature doubles the proficiency bonus for certain ability checks. If a circumstance suggests that your proficiency bonus applies more than once to the same roll, you still add it only once and multiply or divide it only once.

By the same token, if a feature or effect allows you to multiply your proficiency bonus when making an ability check that wouldn’t normally benefit from your proficiency bonus, you still don’t add the bonus to the check. For that check your proficiency bonus is 0, given the fact that multiplying 0 by any number is still 0. For instance, if you lack proficiency in the History skill, you gain no benefit from a feature that lets you double your proficiency bonus when you make Intelligence (History) checks.

So since the expertise doubles the proficiency bonus, but by giving up the proficiency the bonus being applied is now 0 and 0×2=0.