I have a bit of spare time on my hands, and for some reason, stuff like this is fun to me. Here is a list of every single minion that, as the title suggests, could logically be added to Battlegrounds. I may be dedicated, but I am not dedicated enough to tier them. Feel free to comment things I missed, or cards you would like to see added from my list.
Note: I am not adding cards that have anything to do with hand interaction. For example, Don Han'Cho is a type of card I will not be including in this list. Cards that can be added but don't make the most sense will have a * next to them, mainly due to potential to be OP, no text, or inapplicable keywords, IE Rush or Stealth.
Note: This is going to take a long time, so I am going to do the same thing I did with Beasts. A lot of the minions I don't include either have no text, or are completely inapplicable to Battlegrounds. I am only putting asterisks after Inspire minions because it isn't applicable to every Hero.
Murlocs - maybe Murmy, it'd be a good addition. Bilefine spawns taunt, so it should be tt2? The current spawner is fine at tt1.
Demons - weaver should go to tt2, I'm all for witchwood imp. Void ripper can be interesting at tt5.
Mechs - Glow-tron is a tt2 at best, at tt3 it wouldn't be picked over cobalt and annoyer, I think Wargear at tt4 or 5. Micro Mummy is a good idea as well.
Beasts - razormaw is needed at tt3, and I'd add it with king of beasts.
Tribeless - too long and too many, And Tirion is easy pickings right now. Either they add another tribal per tribe or nerf him to 2-cost hero power, but till then, tribeless needs no help. With that said, I want Sally, which will never fly under tt5.
I think that dragons would be cool if only dragon heroes (Neferion, Sindragosa) could get them. The problem with introucing a nrw tribe is getting less of the others offered, like how demons felt before they added two more.
When I get the time, I will make 4 more categories; Elementals, Pirates, Dragons, and Hand-Buff. Like someone touched on earlier, the problem with Dragons and Pirates is that a lot of their effects only work in normal Hearthstone. I honestly expected a bunch of Battlegrounds supporting Dragons from Descent of Dragons to be implemented due to the proximity of the two releases, but that didn't happen.
If you look them over the existing dragons and pirates don't have abilities that lend themselves to the game. They'd have to create mostly new ones to make them viable tribes.
I mean, this isn't wrong. But having a few as essentially boring vanilla minions could be fine. There's a few menagerie tricks they could get, and they'd essentially be non-tribe minions for anyone but Tirion. I don't think that'd be a terrible thing. At very least, it'd be good for the "Beast, Dragon, Murloc" on Menagerie Magician to be fully functional for someone other than Curator.
First off, it doesn't seem impossible to implement stealth. To have something which couldn't be attacked until it attacks would be interesting. Perhaps it couldn't even be attacked if the only minion left, and an enemy would simply not attack, like how a 0-attack egg won't attack. Lotus Assassin would be interesting (good? hrm... dunno... but interesting!) as an anti-token sort of card, but even Jungle Panther might work.
Among potential adds, I really kinda want to see Shattered Sun Cleric. I think it'd be quite interesting to have that around, as an option to buff tribeless minions, but in a reasonable way. Simply having some method for manipulating anything's attack value seems like it has value in the game. I kinda think something like Fungalmancer might easily be too potent. Maybe it'd be acceptable at Tier 5, but not sooner, and it'd still be very strong.
On a quick runthrough Void Terror would be the most insane card to add to the game(and spiderbomb and venomizer are in that "please no" category). Now do the same, but for the adventure specific cards :)
I think Battlegrounds needs some slight mechanical overhauls before it can support much larger minion pools. You have to consider the mathematical practicality of adding new minions into the game mode and how it would impact things like Hero choices in conjunction with their powers.
For example: Jaraxxus could theoretically become unplayable or experience a completely useless hero power. The tavern can logistically only handle 7 minions on it maximum and lower tavern tiers yield less minion choices (for mathematical balancing reasons). More minions means more rolls which means smaller percentages of seeing certain minions and even minion types. You can reach an apex in which an entire tribe may not be seen for a good number of turns and essentially knee-capping any hero focusing on any given tribe, especially if that hero has a hero power that is tied to a specific tribe. Heroes like The Rat King, Jaraxxus, Tirion, etc. all take a significant hit unless you implement new algorithms or mechanics to account for such a large minion pool.
Really, I only comment because there seems to be this common idea that the game mode needs more minions. I think that people who advocate this don't really take into account how this negatively impacts the game mode which kind of already is struggling at times.