Front axle to bars is probably not the best measurement either, you want to look at stack heights.
Assuming you have the same BB-saddle height on both bikes, you want to measure the BB height of the old bike and the bar height of the old bike. Subtract BB height from bar height and you have your ‘stack’ (technically stack is BB to upper headset cup, but BB to bar height is closer to what you want to achieve).
Measure the BB height of the new bike and the bar height of the new bike to get the ‘stack’ of the new bike. Adjust as necessary to achieve same stack as old bike.
A simpler way might be to measure ground – saddle and ground – bars, the difference is the bar ‘drop’ and then you can just replicate that across both bikes.
I’d long forgotten the ‘rules’ about stems and spacers, I’ve been riding lots of spacers and a positive rise stem for ages now, I drop it 5mm occasionally, when I remember, but even on a bike with a very tall headtube, slamming a negative stem would give me a massive saddle – bar drop which is just masochistic for long rides.