You can replace (not renew) your passport irrespective of your expiry date especially when MRP is being issued. 
The current processing time is around 3 months to get the MRP from DC or NY. Looks like you do not have enough time to replace passport from here.

Applying at DC wil not be good option since they take your original passport so if you do not get it on time, you cannot even go to Nepal until you get the passport. 

IF you are from tri-state area, you can apply in NY. NY nepal consulate doesn't take orignal passport (just the copy of it), so until u get ur MRP, you can use ur current passport.  But again if you do not get on time, u shud do visa stamping in ur old passport. So it doesn't help either.

If you are yet to buy the ticket, i suggest you to go to Nepal after 3 months so that you have enough time to get MRP. ....