So I had that problem, that I wanted to play dlc from Mass Effect 2 "Lair of The Shadow Broker". I bought the DLC bundle for ME2 on sale from Origin because it's not available on Steam. Then I launched my ME2 game from Steam which was included in my "EA Play" subscription (which I purchased on Steam too).
I logged into Cerberus Network computer in main menu and I didn't see any way to play DLCs I purchased. I could see the descriptions of all 4 missions that is included in the Bundle but couldn't play them. So I visited Origin Store, looked up my library and saw that DLC Bundle for ME2 is nowhere there but on purchase history on Origin it appears that I bought it. I did my research and found out that with EA Play you get only 2 CD keys instead of 3, which that third is proof that you own the game ("game key"), other ones is "cerberus key" and "promo key".

I looked up my game keys on Steam and saw this:
https://steamuserimages-a.akamaihd.net/ugc/1821118302982126233/CE32C3AD8AA99B6A30E66F0738CD3883834C6F0E/?imw=256&&ima=fit&impolicy=Letterbox&imcolor=%23000000&letterbox=false
There's only 2 codes available.
Funny thing is that in DLCs list it appears I have game key for ME2 but it not appears in settings:
https://steamuserimages-a.akamaihd.net/ugc/1821118302981150475/889CD8B55BEF4A2DD14C352D4A5DC8E382961DEB/?imw=256&&ima=fit&impolicy=Letterbox&imcolor=%23000000&letterbox=false
I wanted to download the DLCs on Origin but you have to own the game to install them. Same thing on Steam.

So basically I have to buy a main game on Steam or Origin and then I can play my DLCs.
If I'm gonna buy it on Steam I need to redeem code on Origin after purchasing ("game key").
If I'm gonna buy it on Origin things get a lot easier because when you install ME2 from Origin it's automatically updates your DLCs with main game.

because you can't then redeem game key from Origin to Steam if you bought it on Origin. You have to buy the game again if you want to play it on Steam.