The traditional solution would be to crank up the master volume to get more power amp overdrive, the only stage common to all channels. So it will of course affect the distortion channels as well, but probably not in a harmful way. However, it of course means you'll be much louder overall1.
Really, that Clean channel seems to be meant to have such a sound that does not make anything "warm", doesn't smooth out any transients, but preserves any details of the sound you give it as much as possible. If you need a more bluesy sound, then this channel is just not the right choice – unless you happen to have a pedal which offers just the right sound, and you don't want the amp to change it in any way.
Otherwise, the natural thing to do is to just use the Rhythm channel instead, in a clean-ish setting: perhaps achieved by attenuating the signal before the amp (guitar volume pot or some rather turned-down pedal). If you also need higher gain scenes on that channel and can't cope with the volume difference, you might even use the lead channel for this, with very much turned-down input. In both cases you can of course also use extra distortion pedals to make up for a bit of the amp's distortion.
But if you want to keep the Rhythm and Lead channels as they are, and are determined to "abuse" the Clean channel as a crunch channel, then you'll have to get that overdrive from somewhere else. Generic tube preamps2 like the ART TubeMP make a good permanent gentle / warm overdrive without altering the frequency response. Again this will affect the other channels as well, but probably not even noticable – only, the signal will have a much higher level into the amp. Just turning down the gain pots (volume in the clean channel) might give exactly the right sound. An opto/tube compressor might be even better.
If you actually want to mod only the clean channel, probably the most effective thing would be to hard-wire such an entire tube preamp into it. You don't really need to know anything about the specs, basically just trace the signal coming from the guitar jack and route it through this extra tube stage. Without an extra tube, I doubt any modification will give you the crunch you're asking for: again, that clean channel was specifically designed to not have any crunch whatsoever!
1It's possible if somewhat ridiculous to "burn away" that extra power with special resistor modules before the cabinets.
2You'll typically find the sold as mic preamps, but they work fine with instrument signals too.