if I mention it during the interview, that might give them a bad
impression, lowering my chances of getting an offer
Then so be it. If they don't consider you as a candidate, then that's what happens.
When should I mention that I'm interested in working part-time?
As early as possible
When you apply, mention that you ideally would be looking for a 3/4/whatever day job. If they don't consider you, then you're not a good fit for the company.
...but be sure to accurately state your requirements
But if you are open to full time, then say that! "3 days a week would be ideal, but I'm open to negotiating". Or similar. It's also useful to say how important it is. "This is a very high priority for me" or "this would be nice, but I'm open to negotiating". They may come back to you and say "Sorry, we aren't willing to negotiate in this area at all." and then you need to decide if that's a deal-breaker or not.
No matter what, be honest!
Don't say nothing and then go through the process (spending their time and yours) and then at the very end say "oh, by the way, part time?". You need to be open about your requirements so that both of you can decide if this is going to work. If you're not open from the beginning, it's just going to be a waste of everyone's time.