Sleep school is sleep training, and success seems to depend on the relative stubborness of the child versus your determination (or capacity to follow through the techniques).

I only know a few people who used sleep school (tresillian) and general reports were it is pretty traumatic and with varying results (one girl I know has 'failed' sleep school three times at almost 2).
Would you consider cosleeping with both kids? You might start getting longer stretches.

I was so sleep deprived by 8 months I weeped over anything. I tried sleep training early on, but you have to consistent and strong and tbh often the desperation for sleep would make me 'give in'. My dd is a particularly stubborn child though and we clash over many things. The moment I gave in and put her in my bed I started feeling human again and more like the mother I wanted to be.

I know sleep school sounds like a magic bullet, but ime there is no one solution to sleep issues. You need more support to help deal with your toddler and maybe sleeps in the day. That and cosleeping will get you through the next few months, but obviously it brings it's own challenges.

Hth. I am worried about the new bub being as wakeful as dd, because I really struggled there for a while.