Well if he will give you more after the ten days, that is a solution. If you can manage at least 4 weeks you should be okay.
Hopefully he will realize that just because symptoms go away, that doesn't mean the bacteria is dead.
It's just like humans, if you have ever had an infection and get antibiotics to treat it, they always say "Take the FULL prescription, do not stop taking the medication early just because you feel better!" It's the same for rats, they may look fine and seem better for a few days, maybe even a couple weeks, but then the infection rears it head and this is how some rats develop serious problems like pneumonia.
You could call and say you spilled it and get more to keep on hand?
Hopefully he is dosing him correctly too. If you post Tyrion's weight and the strength of the Baytril I can check the dose.
He should really be on Baytril AND Doxy as of this point.
Unfortunately there are definitely some vets who play this game of making money off of revisits and refills on prescriptions.
You can try getting the antibiotics online, although I don't know what that is like in the UK as I am in the US.
This is where I get Baytril:
They do ship internationally, although I do not know how expensive or how long it may be.