The effect of side-chaining would depend on the compressor.
If it's a full-range compressor (i.e. not a multiband) it will apply compression to the entire track the compressor is on every time it received a side-chain signal.
A multiband compressor with a switchable sidechain for each band would only compress in the frequency bands with the sidechain activated. There actually aren't many compressors with per-band sidechaining, Waves C6 is the one I use. Compressor pumping can be an issue though, so it's best done subtly.
An alternative is careful use of a multiband comp without side-chaining just reducing the hat's frequency band. It would require quite a bit of automation to set that up though, unless compressing everything in that frequency band throughout is OK.
Careful eq notching can help as well. R-mix is another possibility if you have it and it will work for you (it point blank refuses to load in SPlat for me, I think it,s only compatible with X2). Ozone can do this sort of thing as well.
To be honest, if you have the original drum tracks the easiest and least obtrusive thing to do is simply re-mix them.