I think both are a little too soft for me. The HL5 is a soft feeling blade with a lot of flex, so at least my preference is to pair it with harder rubbers.
Specifically, the T05FX played super well with the Viscaria. I felt that with a little flick I can make the ball go anywhere I want with good pace. So that was the first rubber I tried on the BH and it just didn't feel good. When I try to do quick counters mostly borrowing my opponent's speed/spin, it doesn't feel very crisp, I didn't feel like I was in control of the shot. When I use more power it feels like I'm slapping the ball with a ribbon.
The H8-80 37 degrees was better. It was my first time using that rubber and it was surprisingly light and soft for a DHS rubber. It felt way softer than my 37 degree H3. It didn't feel as mushy as the T05FX, but it just felt a bit too slow. Even when I put more power into it it still felt slow. It wasn't like the D05 or the D09C which aren't super quick either with lower power strokes but can really pick up some speed with a harder swing. I think it's a good control rubber for the BH side, so may fit certain styles.
BTW, I tried the T05 after the T05FX. I really liked the T05FX on the Viscaria and I used the T05 on a TB ALC for a couple years so I really wanted to give the T05 family another try. I thought the harder sponge of the T05 would take care of the T05FX's problem, but it really didn't. I still felt a lack of control except it's now going to the wrong place faster.
The rubber I felt the best on the BH side, even better than the D05, was actually the D09C. When I twiddle I really loved how it felt, I just don't have a red sheet but I have 2 sheets of the D05 so I went with the best available. I've never tried the K3, but I've read that it's somewhat comparable to the D09C. Since you liked the K3 on your FH side, have you tried it on your BH?