Myself and another owner agreed to a trade and is awaiting approval by other owners.

However, the other owner is over his budget for next season by $1.2M and he is concerned the trade won't go through (he is slightly under budget for the current season). I think it will since the budget overage is for next season and is only a projection anyway.

Anyone have any insight on this?

If a team is over the allowed budget in any season (current or future), the trade will be nullified. 

Yup. Trade nullified.

Sounds like he couldn't make the trade in the first place. Not cool.

Those future budgets aren't projections-- they're strict limits on how much future salary you can take on.  You can't take on more than 80% of your current budget for next season (with further decrements for farther away seasons).  This is done in order to limit the number of really ridiculous long-term contracts an owner can give, because the owner could then walk away from a nightmare of a franchise.

Is there anything an owner can do to increase a future year's player salary during the season?

Cash in trades does not do it.  Budget transfers into payroll I think might... anyone remember that?