One (or more) of several possible issues could be happening for you.
The most likely issue is that you do not have room in your BOff Candidate Roster. Your BOff candidate roster can hold a maximum of 12 BOff candidates. If you do have 12 BOff candidates, either refuse 1 or 2 of them, or mail up to five of them to yourself. Then try to buy a BOff off the exchange.
Here are a couple other potential issues that could be affecting you:
1) No room in your inventory - I don't know why this is the case, but several in-game items spend a split-second in your inventory before being redirected to where they are supposed to go. For example, any mission that awards a Dilithium Ore Container will require you to have inventory space to temporarily store the Container before it is automatically (and immediately) opened for you. I believe BOffs are treated similarly.
2) The BOff you are trying to buy has already been bought by someone else - When Season 6 came out, the exchange was severely out-of-sync. Sold items would continue to show up in exchange searches for an hour or longer. Trying to buy them would give you the generic "You cannot purchase this item"-type messages. However, I haven't seen this problem reported with Season 7.
If the above do not resolve your difficulty, please post the exact error message you are getting when trying to buy a BOff off the exchange. That could narrow down the possible root causes.
For kicks, I bought an opposing faction BOff just to see if the game would allow it. The game does allow such a thing to occur. The opposing faction BOff appeared in my BOff Candidate roster. The BOff even came with a bright red warning label telling me I can't use it.