Hey there, welcome to the forum! I'm not sure if you're still checking this post or even still have the Heartbreaker, but for what it's worth I've tried EL-34, 6L6 and 6V6 tubes in my Heartbreaker and stuck with the 6V6's. I feel that the 6V6 tubes kind of deliver a good Fender-ish vibe to the amp, but still can be driven to some pretty decent rock and hard rock tones when you spend some time playing with the controls and adding an external fuzz/overdrive/distortion pedal. I will say that I liked EL-34's a lot better with my Gibson SG and 6L6's for my PRS Custom 24, but my Fender AVRI '62 Strat responds best to the 6V6's. You will NOT get a metal tone without mods to the amp or a pedal (I use a Voodoo Labs Sparkle Drive).
It is a challenging amp in that every single control seems to have a strong effect on your sound, and the amp is inherently bassy to start with. The only drawback that I see to 6V6's in this amp is that some say you should only use the Tweed Variac option when running these tubes. I don't fully understand the reasoning behind this (I'm not very technical-minded) but there are those on this forum that know the Heartbreaker very well and could help you out with those kind of specs. Monsta Tone is a big Heartbreaker fan and offers some pretty cool mods out of his repair shop for it, so he'd be someone to contact with questions or repair/mod work.