Hm... never have had a hard time getting a nice warm, yet biting and clear clean tone out of my Stratocaster..... haha!!

To relate somewhat... I do have a Strat that I custom built into a double fat Strat.... and did some other mods to make it sound more Les Paul'ish... but I have a set of JB/'59s in it.... with one tone knob and one volume knob... I find that my bridge pickup is constantly warm as well... so what I normally do is turn the tone knob all the way to the treble side or atleast half... depending on how warm/bright I want it to sound... and play with through both pickups. This gives me a nice full sound. I try to get my sound dialed in with as little help from the amp as possible... that way when I change amps... the tone is relatively the same. String choice might also help you get some more warmth out of your bridge pickup... I use GHS ZW Boomers .10's. They sound awesome through clean settings... as well as distorted ones. They probably contribute the most character to the tone of my guitar out of any strings I've ever played... But yeah... just see what works for you