There are better ways to shorten your workouts than cutting rest time short. Definitely make sure you're getting adequate rest between heavy sets. Don't know what space/equipment you have available but there are some good tips here to speed things up.
What To Do If Workouts Need To Be Shorter | Mark Rippetoe
Not sure what to advise on press except that I believe sets of 5 are what drive progress. 4s quickly turn into 3s, then 2s, then 1s, and then you're stuck. If you need a small reset and then 1lb increments to keep moving forward with sets of 5 then that's something to consider.