There's a few ways to make it easier on yourself though.
Firstly, you can wait for an event that increases stability, and then before you click accept, click westernise, though this is a bit of a gamey solution. Religious turmoil can actually be good for this because solving it gives you +2 stability, and the deal with rebels mission gives you +1 stability, though those might be rather hard to do if you're in the middle of westernising.
Secondly, you can wait for a good admin monarch, and take a good admin advisor, the other forms of monarch points won't matter much.
Thirdly, if possible you can just wait for yourself to accumulate 900 admin monarch points, then instantly go from -3 to 0 stability. This will save you from having to take all those -1 stability or lose a bunch of points and gain a rebellion events.
One other thing I should warn about is the issues I wound up running into when westernising in my recent Golden Horde game. I tried to do it as soon as possibly by snuggling up to Genoa's black sea provinces. This worked, and I eventually managed to westernise by about 1500, but I quickly realised this might not have been the best strategy when the war I was in with Muscowy when I finished quickly went bad very quickly. This was a war that was going alright up until that point, Moscow may have had more units, and a better quality of military, but with good leadership and tactics I was managing to hold my own.
Then I westernised and my units all automatically swapped to western types, which was bad for two reasons. The first was that all of them dropped to 0 morale, which froze them in place and allowed Moscow to instantly kill some 18 regiments of cavalry, and the entire rest of my army.
The second is that equivelant western units were actually worse. The nomad tech group starts the game with slightly better units than the western tech group, this only evens out at about tech level 10, which was something I was nowhere near close to having (this is ignoring all the little discipline and shock bonuses you get along the way).
Even after the war was over, the other nomads were kicking my butt as well on account of having better units. I eventually just threw my arms up and gave up after Moscow declared war again.