There are so many rubbers you could use for your BH. Sometimes what you get for BH is related, to some extent to:
1) how you play (pushing, blocking, looping),
2) level (you said you are intermediate but one person's intermediate could be another person's beginner: I know people who feel they are intermediate/advanced players but they have only played basement players and as soon as they play club/league/tournament level players, they realize they are really beginners) and
3) what you want: do you want fast, slow, spinny, high spin/speed ratio, high speed/spin ratio, etc. And these choices, hopefully would have something to do with level and play style as well.
All that being said, a medium or medium soft Euro tensor or Japanese grippy/non tacky rubber is usually the most useful thing for a BH rubber. And most of the major non Chinese brands have several versions that would be good.
Examples would be:
T05fx, T80fx, T64fx
EL-P, FX-P
Omega V Europe, Omega IV Europe, Vega Europe
Baracuda
Aurus Soft
Rakza 7 Soft
The list goes on.
Some people prefer Chinese rubbers for BH. Not many but recently there has been a new market for this with the 40+ Poly Ball.
So a tacky rubber like H3 with a softer sponge, can work. Or some of the semi-tacky Chinese rubbers with softer springier sponges that are sort of imitations of Tensor rubbers but with less spring and more tack, those work well too.
The question really is, how do you play and what do you want.
At some point someone is going to say: "Video footage of you playing will help." Maybe it would. Or you could just make a list of all the rubbers you are considering, make a cue card for each rubber, put them on a dart board and close your eyes and throw till a dart sticks in a cue card and that's your rubber.
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