I'm surprised to hear that the BAT is warm sounding with the Focals. Even though the term warm is subjective, I would be inclined to look elsewhere for that particular issue.
I tend to agree with you that more power (especially in the form of current) will be welcome. I'll leave it for others to recommend solid state alternatives, but I would think that the Bat tube monoblocks, coupled with one of their companion preamps would work well with your speakers. I cant say how they will mesh with the Rhea. Also, keep in mind that you have some high quality components there, the performance of which can be heavily influenced by your choices in ancillary equipment such as isolation devices, racks, power cords, cables, interconnects, etc.
One last thing, don't conclude that separates are an automatic upgrade. There are some very competitive integrateds out there.