The Athletic is reporting that Jordi Alba won't take up one of the DP slots:

Messi and Busquets will occupy two of the club’s three DP spots. Leoandro Campana or Rodolfo Pizarro will occupy the third.

Inter Miami actually had three DPs already before the Messi deal was announced: Gregore, Rodolfo Pizarro and Leonardo Campana. Inter Miami also has a history of salary cap violations and being fined for paying players outside of the cap (they were functionally carrying 4 DPs by compensating Blaise Matuidi without reporting it). With that said, there have been a number of head-scratching moves in MLS historically where players have seemingly signed for much less than their value (see Gareth Bale to LAFC), and the arrangents for these contracts are not very transparent.

For Miami, It seems like it will probably need to be some combination of moving their existing DPs and using accounting mechanisms to reclassify existing DPs. MLS has a complicated "allocation money" system wherein teams receive general allocation money (GAM) and targeted allocation money (TAM). This money can be traded, eg in return for additional international spots or players, and Inter Miami has accumulated a good bit over the last several years. (Their sporting director, Chris Henderson, used to work for the Sounders and is very good at his job).

Miami can use this allocation money to buy down the cap hit of players so that they are no longer DP players, but instead are "TAM-level" players (players who make more than the maximum salary for non-DP players under salary cap rules, but less than a DP - they can be paid with targeted allocation money). That would free up a DP slot.

My guess is that Miami will probably use allocation money to move Gregore to a TAM position, as he has the lowest salary of their current DPs. They may try to trade Campana and/or buy out Pizarro (who doesn't have much time left on his contract) to free up the other. Then the three DPs are Messi, Busquets, and Campana/Pizarro. A Miami fan may have a better idea. The Athletic also had a good rundown.

How are they getting Alba on a non-DP contract? I have no idea. With that said, my guess is that MLS is very willing to get creative with some of the rule interpret to make this Miami-Barcelona reunion happen, and because of the single entity nature of the league most other teams (and many of us fans) won't really care.