Well I've had depression alongside other mental health problems and taken malaria tablets. I didn't notice anything or even know about it. I would say speak to the doctor. It will all be confidential anyway and they will be able to give you an informed opinion. They may also be able to put you in touch with professionals who can help you should you feel like you need it. If you really feel you can't go to the doctor the NHS helpline could be a useful source.
Well done by the way. It sounds like you've come a long way. I hope you continue to feel better. Remember depression affects a surprising number of people and you are not alone in this. It can happen for a variety of reasons or for none at all. There is NOTHING to be ashamed about (I know this is easier said than done).
The fact that you are worried may be, in itself, a cause for concern because, even if the tablets weren't to affect you, you may perceive them to have an effect (the placebo effect) so some reassurance is probably needed.
If you do need the more expensive tablets and don't want to tell your dad you could go and get them yourself if he gave you the money. You'd have to pay the difference but there is a way out.
Good luck and take care xxx