Who says that if you start benching you will get slower, this is a common misconception.

Start doing weights with low reps, high weight. ONLY big compound movements like bench press, deadlift, squat and military press.
You go to the gym, do that for 5 reps for 5 sets, at the heaviest weight you can twice a week, and you will put on weight. Start eating more with this regime and the extra muscle will metabolise the fat you should be putting on.

Its a common old-school-trainer mistake that weights slow you down. They do, if done incorrectly. Do whats descrbied above and you may even see an improvement in your boxing.