Your population looks fine to me at 136 residents, but that's quite a bubble you have around teens. Before we had the multiple entrance mod, I would say that my schools maxed out at around 25, maybe almost 30 students each but it did take them way too long to enter and exit without some help even with no other obvious drains on things. You have 45 children/teen residents, so assuming they are all trying to cram themselves into the same school together with some adults who probably work there then yes that's a bit much with or without the mod.

So definitely would agree with an additional school. You might consider an elementary and a high school, it's a bit fussy to set up but some of us think that's more realistic than distance based. It doesn't look like this bubble is growing or even sustainable judging from the numbers of toddlers, babies, and pregnancies, at least not right away, so you many not need more than two but the Careers mod specialty schools can also help if you still feel there's an overflow on the teen end with 31.

Sims need to have a reason to register for a new school. To get them to do that automatically to the one that is closer to their house, you would have to bulldoze the original school and replace it with a copy. Same applies if you want them to follow SP Lot Options settings (GoHere mod presence also required) to coax them into registering for elementary vs. high schools or any other system of separation you may have. Sometimes it takes a few tries before they get the idea.

Alternatively, you could shove the sims into the school you want them to attend with MasterController.
NRaas > MC > (Sim) > Intermediate > Career > Choose School

Note, as you have already seen, that no matter how it's done the sims who end up at a different school (from the bulldoze replacement or MC) will start over at 0 with their scholastic performance. You can reset those one by one as well with MC, though it's a bit tedious. Sometimes when I've had to do this, I've only bothered resetting the ones who had scores of over 60 or so, plus any who might have been doing not quite as good but who were about to age up soon. The extra school work to catch back up again builds character (up to a point).
NRaas > MC > (Sim) > Intermediate > Career > School Performance > (-100 to 100)