I like some of the already existing card choices you have here, especially Al'Akir and Bog Slosher. I think Bog Slosher should probably go up a tier, though. Soldier of Fortune, however, I'd say is like a Tier 0. It's absolutely not worth having a stat stick early on with that big of a drawback. As for your custom cards, there are a few problems, discussed below.
You have a few themes, Windfury, start of turn, selling Elementals, and Flame Elemental. Out of these four:
Windfury is good, but not enough support. Out of all these tactics, this is probably the one I'd like to see added in to the Battlegrounds. It can scale very well into mid and late game, especially with Divine Shield and it has been proven to be effective already with Murlocs.
Start of turn, too gimmicky and niche. Start of turn does not scale well at all, and your 6 mana 5/5 that double start of turn effects will only affect perhaps 6 cards (including three Hero Powers, I do not know if you took that into consideration). So, only 9 cards in the entire Battlegrounds pool, and none of these buffs scale. That's awful. This kind of card is like a Brann/Rivendare, but is a whole lot worse *and* it's in Tier 6 unlike them, both in Tier 5.
Selling Elementals, far too weak. Look at Malygos. He procs off any Battlecry. You probably need to rework the tiering of these two and make them far easier to activate, or have a much bigger payout.
Flame Elemental, very hard to judge. The only thing we have to help balance are Primalfin and Muruzond. I hesitate to add an easy way to give triples.
If I were you, I'd look at a few cards such as Cauldron Elemental (Tier 4 or 5), Whirlwind Tempest (Tier 6, the Megasaur of Elementals), Siamat (Tier 5), Mountain Giant could be *very* *very* interesting, as it rewards player for floating cards in their hand, something that not everyone can do all the time. No idea what tier Mountain Giant could be, though in honesty, the idea of it in battlegrounds is a little far-fetched.
Thanks for the feedback and for your time! I agree with you on a lot of points. I think it is important to have multiple themes to keep the game varied, and I understand the concerns with some of them.
Start of turn not scaling well: I was aware of this, at least for the stat buffers. I think it is important that you can't keep early tier minions for the entire game. and that you transition from an early/ midgame strategy to a lategame strategy. The actual lategame cards in my opinion are the divine shield, poisonous and doubling ones. With one doubling, two poisonous givers, a divine shield giver and the elemental that has everything, you get 5 poisonous divine shield elementals which is pretty decent for endgame. I absolutely understand the concern but I would love to test it out like this. (Which is of course not going to happen)
Bog Slosher I had it at tier 3. The sole reason I put it at 2 is because I had too much at 3 and too little at 2. I believed it was only slightly too good for tier 2. Compared to Nathrezim Overseer. But it buffs everything so I see where I'm wrong at doing so.
Soldier of Fortune. I know you are right about this, I just find the card to be hilarious in battlegrounds. Maybe at tier 1 as a trap?
I think I am going to rework the selling elementals ones. I wanted more synergy with the flame elementals, which I think are a great addition though they need very careful balance. It is kind of like the golden kadgar or kadgar bran + tokens combo. You have to invest some health/ take some risk for the payout. I love strategies like these even though they lead to losses most of the time.
About your suggestions.
Siamat was considered and I still like it, but rush (like overload) has no effect in battlegrounds so I didn't find the card the most elegant choice.
Cauldron Elemental is a great option, but it is definitely not tier 5 in my mind. You really want game winners at that stage. Though it does seem slightly too powerful for 4 as well. Still I think it belongs on 4.
Whirlwind Tempest would be a tier 5 card for me. I don't think it is on mama bear/ kalecgos/ megasaur level. You really need a lot of stuff to fall into place for it to do much.
I don't think the interaction of mountain giant would be immediately understood in the context of battleground. Would it only cost 2 gold if you are floating a card? etc? In that case it seems like tier 4 to me.
Again thanks for the feedback!
EDIT: I made some adjustments, but now it seems a bit one-dimensional