I haven't seen anything on how much influence a doctor level makes to the injuries. It would be pretty hard to determine a reduction in the duration of injuries since the only information you are given is how many weeks.
If I was in your position, I would probably hire a level 3 doctor. That should help reduce your injury time, and the impact to your economy would be very little. The remaining money I would look at maybe upgrading to a level 6 trainer to improve on training speed. That is assuming that you don't need any help on your roster or I would save up to find a player on the TL. That's my 2 cents...