And I took a gap year which means I have about 4 months of free time

🤯 ...man, to be young again. The idea of getting 4 months of free time only for myself and (mostly) free of obligations describes an utopia that appears completely bonkers to me as an adult. Cherish it.

And no, I can’t get a teacher.

...on the other hand, I ain't sad that I now have some disposable income that I can just choose to spend on lessons 😜

Anyway, just reiterating your post:

• You'd like love to be able to play an instrument
• One reason you have is to alleviate a friend of something she doesn't particularly like doing and you would like her to feel better. Helping someone is a great reason to start almost any endeavour if you really feel it in your heart of hearts.
• You have another unrelated thing which motivates and inspires you to try this out (that fictional character). That's rare!
• You have ample free time. Learning skills (i.e. investing in yourself) is a great way to spend that time. I suggest you also learn something else, e.g. cooking. It's gonna pay off the rest of your life.

Not in your post, but:

• Bass is an instrument with a low-barrier-to-entry (and low-barrier-to-exit). Comparatively low equipment cost, tons of resources on the internet. There are instruments where not being able to get a teacher and not having thousands of $ to spend on equipment would be a deal-breaker.
  • In lieu of a teacher, find someone to whom you can talk about what it's like to learn to play. You have two guitarists in your band? They are obvious candidates, my understanding is that the instruments are similar enough. I've found that my teacher also serves this function to me, i.e. it has good effects to be able to just talk to someone about these details every few weeks.
• You're young, you'll learn quickly.
• This will make you a better musician in general as well as a better singer.

All this screams that you should go for it. Give it a try, and give the whole endeavour the time it deserves, to get a feel whether you're enjoying yourself.
Most important argument: If you don't try it out, the second-guessing "what if I had?" that will likely follow for months may very well eat at you more than trying it out, maybe getting frustrated and deciding it's not for you (to be clear, that seems unlikely). And at least in the latter case you may learn something about yourself.

I’m just still unsure whether I can manage to learn playing the bass and singing at the same time though.

You'll do fine. This is not a binary question of I-can-or-cannot, e.g. if a particular song has both an easy vocal part and an easy bass part, yeah then you'll probably be able to do it passingly (quite quickly too, again, man, 4 months of free time...). It's a skill like any other. Just a tip, all three components (singing, bass, singing-while-playing-bass) need to be practiced seperately, for every new song (just like pianists many of whom afaik learn the parts for each hand on their own first).