Thank you for your replies.
Update on the ticket:
The support team member is still persisting that he/she can fix my problem if I supply my password.
Snip of ticket:
"I understand your restraint. What you described is a known problem and we are working on it.
Up to now, I tried it a couple of times with the clients account and the order came through.
You will receive an order confirmation, so you can check what I have done.
Also if you like, you can change the password after.
I will not use our password to harm you or what so ever, just to help you."
I don't think the support team member could do anything malicious with my account aside from changing a few profile values (no order is completed yet, very limited personal information in it)
Opinions?