It's kind of hard to answer that without knowing what other teams are offering. The Bruins farm system may not be deep but it isn't totally barren, they could cobble together an extremely competitive package if they wanted to.
The Bruins 1st is going to be 28-32 but I don't think it's that big of a deal. Nobody in the lottery range is going to deal their pick in the Bedard sweepstakes unprotected. The other teams in the mix are probably in the 20's too. It's not a massive difference the Bruins couldn't make up elsewhere.
Yes, I think that they could put together a competitive offer to AZ, but their only assets with any significant value (who aren't already rostered) are really limited to just Lohrei, Lysell, and Merkulov (maybe Beecher, I guess, but I'm not counting on him) and draft picks (the latter of which every team has), so they have to hope that some other team doesn't have and isn't willing to deal assets that are more attractive to AZ. And to be honest, how sold are we on Lysell? Small sample size, but he stunk in the recent WJC and has been only okay in the AHL (I realize that he' just turned 20, but to me he's a
very good, but not great, prospect).
Also, and this is important, the Bs are going to have to burn one or more of their few assets just to get AZ (or some other team) to eat a bad contract, so it wouldn't be just Chychrun that they'd need to buy, but instead Chychrun and say, Reilly's contract.
I don't think that Bs are prohibited from acquiring Chychrun, I just think that they have more hurdles than do some other teams (not saying that other teams don't have hurdles).