Very good point Steve30.
Make sure you take the wheels to an experienced shop that AT LEAST has a Hunter Road Force machine. To properly balance the tires you'd need at least the Hunter Road force, if people who use that machine have some brain with them , it's even better .
Get the tires remounted and rebalaced properly. Why remounted? well I can almost guarantee that they're mounted improperly right now . When you go to remount and rebalance them ask the installer what the Yellow and Red dots are for on the tire. If he cant answer that question then call him an idiot and go elsewhere.
I got a question for you, what RPM are you at when doing 100-120? Are you absolutely sure that the shake is only dependent on the speed and NOT the RPM . If the shake is only based on speed then its guaranteed that its your rims/tires.
some shitty china wheels cant be balanced even when new , because they're shitty.
Some tires cant be balanced becaseu they're shitty even when new.
Sometimes the tire will look perfectly fine but will have a broken belt in it , mostly happens to old tires. So check the date on your tires, if older then 5 years, then throw them in the garbage
E46s are extremely sensitive to anything that has to do with suspension , tires included. E46s dont like china parts and usually reject them by making all sorts of noises and vibrations. Check your car for the presence of crappy china parts and replace with quality parts