Can a warlock use Pact of the Blade to restructure a magical pact weapon into a different form?

I don’t have the exact text, but I’m going to be running a Fey Warlock for an upcoming campaign. I remember this for Pact of the Blade at level 3: You may create your pact weapon in any form you wish, and you have proficiency with your weapon.

I don’t have the exact wording, but I know that you can make a magical weapon your pact weapon by spending some time with it, which can be done as part of a short rest.

My question is:

Let’s say the party finds a magical axe that let’s say its a +1 axe. If I make that weapon my pact weapon, then I summon it: can I choose what form it comes in? For instance, taking a magical axe, making it my pact weapon, and then reforming it as, a rapier instead. Or finding a magical weapon with an elemental-alignment, and restructuring it into a weapon better suited for the character?

The only time I could think that you would consider trying this with the Pact of the Blade is if you’re running a dex-based magical warrior, but it was an interesting idea that popped into my head. Does this work?