Backstrom on LTIR means they will have like 2.9m in cap space. Which won’t grow unless they either change out some players, or Oshie also goes on LTIR.
They have all year to fill their LTIR space. And that 2.9m will go farther as the season goes, I believe (as players play the season, their cap cost drops in percentages along with how much season has been played (also in percentages).
So I believe they could fit a 5.8m player into that 2.9m a lot, if the season is 50% done.
And that’s still in laying Oshie as active and playing.
Unfortunately, you can only accrue cap space if you are not using LTIR. We'll be in it all year, so there's no pro-rating cap magic we can pull off. LTIR also works technically not by "subtracting" the player's cap hit but by "adding" to the cap limit the amount over the Cap you are. Before putting Backstrom's 9.2m on LTIR, you want to get as close to 9.2m over the cap as possible to maximize that bonus space.
If Oshie is good to go, I assume they send Bear down. If he passes through, then they save $1.1m and are $9.12m over the cap, almost maximizing Backstrom's $9.2m. If he's claimed, my guess is they then call up Miro to put them at $9.16m over, once again almost maximizing Backstrom's $9.2m.
If Oshie isn't then you have another $5.75M to play with. If you don't want to have to send Bear down before the season through waivers though, you want to use the ~$5m right away otherwise you're "wasting" the vast majority of Oshie's LTIR pool. My guess in this scenario would still be that they waive Bear and keep Oshie on LTI until either he's good to go or they can make an in-season addition.