The most practical solution is to use the two internal pitchers coming off injury in the bullpen in Eovaldi and Wright, to get more IP from the starters, and to stop using the back end of the bullpen in the rotation.
It doesn’t look like either Wright or Eovaldi are going to be capable of handling starter duty this year anyway, so the bullpen is the best and really only option for them. Both can certainly be very effective pitchers picking up about maybe 60-80 IP combined in relief. This would mitigate th need for using the largely questionable Brasier in high leverage situations and mitigate using Barnes in innings only started by the better hitters in each lineup as well as in consecutive games.
But the onus of improvement needs to rest with the rotation. If this rotation doesn’t do their job, then the entire bullpen would need an overhaul top to bottoms for multiple reasons.
If Eovaldi closes, I’m not worried. Closers largely come in for clean ninth inning work nowadays. Eovaldi is an established veteran MLB pitcher with some excellent pitches who should be able to give the Sox 1 IP without issue. The biggest fear with flame-throwing closers is always overthrowing themselves into injury, which Eovaldi has done a few times in his career, but always as a starter.
Wright has some issues, but also has a better track record for success than the entire Sox pen already and at the very least represents a solid change of pace.
But getting a fifth starter is paramount here as opposed to using the bullpen to handle all 9 IP every fifth day after 4 straight games of pitching 3-4 IP. That is just begging for issues in August...