There are ways to get stuff done later in the season. For example, you trade a young guy and cash to another team and then you sign the cut vet you want for the prorated minimum.
There is also the possibility of signing guys like Joseph and Christmas to partially guaranteed contracts where you can cut them in December if needed at no cost.
I wouldn't worry.
While it's true there are ways, they are fewer than you think. For instance, the contract for Smith is already signed -- he gets $25k for coming to camp, and then it becomes guaranteed for the year if he makes the team. I bet Christmas' contract is similar, but with a little more paid as a guarantee, probably $75k. He didn't merely get a camp invite -- he's signed a contract that already includes either a partial or full guarantee if he makes the team (and I'm betting on a full guarantee.) Joseph, from what I've seen reported, is non-guaranteed until the end of training camp. If he makes the team, he's guaranteed for the year, and is non-guaranteed again next off-season through some date.
Additionally, there is a limit as to how much cash you can trade during a season. They've already used a good chunk (I've never seen the exact amount, but I'd guess around half) in the Courtney Lee deal, to pay off Portland. So, yes, you can ship off a rookie, or whichever vet making the min you've found not useful, to create an opening for another vet at the min, but you're probably going to have to include a draft pick of some sort to get it done.
If the numbers are correct (and I do think that's a big if -- I'd love to see them sourced, as opposed to ESPN sourcing using ShamSports as a source and then a third-party saying "the same number is on two sites, it must be good") the Celtics will be very, very limited in their roster flexibility. That could be by design, as the front office may not see anyone likely to be worth getting becoming available mid-season. (For instance, you may not think Brandon Bass or Jeff Green are starting big man quality, but who are you going to get for the vet min in the middle of the season that is better?) I wouldn't be surprised to see them throw whatever small amount of cash they have left at some vet as a similar "come-to-training-camp-and-earn-a-spot-on-the-team bonus" to give them one other player to look at for a final roster spot. I also wouldn't be surprised to see them start the season with only 14 guaranteed contracts if 2 of Smith, Joseph, and Christmas just don't look like NBA players during camp.
I'm not personally worried about the lack of flexibility, but it is a real issue they're going to have to deal with this year. Next year I think they're going to have no problem with the hard cap at all, but this year will be a struggle.