What visa to be employed by a company while completing a PhD as a foreign student in the US?
If you are going to be employed full-time (>20 hours week), you will not be able to use F-1 without CPT/OPT authorization.

Since you are still working on your degree, OPT will generally not allow you to work more than 20 hours per week (during school in season). CPT authorization can technically give you the full-time off-campus employment option; however, you are supposed to actually do work for the company, not finish your degree requirements per se. The International Student Office of your university will help you to determine if you are eligible for CPT and figure out if your plan works. 

I would consider different options:


be employed by your company in your home country and consider the L-visa for intra-company transfers
get OPT while on F-1 but work part-time for double the salary if that is a viable option for your company


Changing to J-visa has its own consequences and I am not qualified enough to discuss them. In any case, you should consult a lawyer at your company (who are supposedly the ones doing all the heavy-lifting for you) or at least some external immigration lawyer.