Tweed Bassmans are heavy mainly because of the transformers and 4x10" speaker cab. It was a bass amp, so they used larger iron set. Larger than their 2 x 5881 guitar amps.
I don't know of any 15w to 20w guitar amp that has more 'head room' than a DR.

You might try looking at old Ampeg combo amps? They were clean. Some of them used different power tubes than 6V6/6L6, but they make some of those tubes again.
How much more clean volume do you think you need? Some guys find that they can get enough of a little more clean from an amp with a
very efficient speaker. Or if they use 2 very efficient speakers.
They make oversized OT's for many tube amps, they make them for DR's. They stay cleaner longer, but if your tubes are distorting, a larger OT probably won't help. Hard to know until you try it. An oversized OT in my little Champ made a difference.
With both a 12" (or 2 x 12") efficient speaker(s) and an oversized OT, it might be enough, but it will cost you to find out.
Other than that, to get more clean you need more power. And more power means larger iron set with bigger power tubes.
Fender made a tweed Pro amp that was ~30w, 2 x 5881's, with a single 15". Might be an option? Fenders next up was 35w amps with either 1 x 12" or 2 x 10" speakers. The iron set in those is not too bad in size and weight.