Will Doxycycline work again or as well for LPP if I had to stop treatment for a few months?

In my opinion, the doxycycline will help provided that the condition has not worsened. I would like to recommend that you lower your dose to doxycycline 40 mg a day or 20 mg twice a day. In addition, minoxidil has shown the highest rate of response for patients with lppPRPHAIRMD.COM 

Yes, this will most likely give you improvement again, but you may also experience the same side effects that caused you to discontinue use. I suggest seeing a hair loss expert now to evaluate and go over your condition and provide you information about your best treatment options. There are great non-invasive options like prp/progesterone and/or hair transplantation. See an expert. Best, Dr. Emer

What you have read about treatments not working if you start and stop does not really apply to doxycycline. Provided your LPP has the same activity as it did before, it will likely help. However, you may get the same side effects so speak to your dermatologist about what is the best steps to take. 

The discussion on the use of Doxycycline instead of how you are now being treated is a discussion you must have with your treating physician