You get 1750 Guantlet Gems a day from doing the Gauntlet if you are clearing it. Double that if you are resetting. Use those and sink them all into Dog and store resets and you get a minimum of 15 Dog frags a day, (or 20 every 3rd day or so)
In comparison with PVP you can only get a max of 10 frags per hero, and that’s totally random. Some days you will get 3. Matador gives another 3 from daily quests. You can reset PVP, but one resets and best case still means you spent 2100 PVP gems for 20 hero fragments (or 23 in Matadors case). Worst case you are looking at 6 frags with a reset for 700 gems, though you could reset another time before the price jumps.
This should be a no brainer. Dump everything into Dog as he is not only a top tier in both PVP AND PVE, but he is easiest to get heronium from (unless you really want Maven and/or Flatline since they have been added to the Guantlet store).
After you get Dog I would focus on Mandrake as he is the only really good hero choice from the PVP store. The rest of the heroes are pretty sub par. If you are high enough you can also get an additional 9 Mandrake frags from the hard missions, bringing the minimum total up to 12 with no PVP reset, or max 19 a day.
Once you get a few 10* heroes it becomes much easier to start funneling everything into other heroes you want leveled up. For instance you can get 15-20 frags from Dog and another 12-19 from Mandrake once you have them maxed, making for 27-39 frags a day from heronium. Add in resets and this number could nearly double, going to 54-78 a day from just two heroes. Whatever you do just make sure you have a good source in the Gauntlet store and at least one hero you focus on in PVP every day.