Question

Would it be wise for Tesla to sell batteries to other automakers at a premium or, considering they are supply constrained, just keep them for use in their own cars?

Answer 1

No, Tesla shouldn't sell their batteries and should use it in their own cars.
As of now there is no other Electric Car companies which could really compete against Tesla and they have good sales, and maybe if they start selling their batteries already while being supply constrained, some customers might as well lose patience and probably go for a different decision.
Selling their batteries would also give other Electric Car companies an opportunity to try Tesla's batteries and that is not something Tesla wants to do while being supply constrained. It is not good to let other competitors enter your market easily.

But in the meanwhile if they decided to sell their batteries, they actually might get the full batteries market under them because of their popularity, affordablity and success history but they should be able to manage bring an end for their supply constraints.
Considering the current market conditions affected by Corona, there is a huge decline in Cars industry and this is a good reason for them to start selling their batteries now. If Tesla decides to sell,then they should definitely have a strong plan on putting an end to their supply constraints.

September 22nd 2020, Tesla is expected to release some major updates regarding their battery technology. Elon Musk have been hyping this event since Tesla’s Q1 2020 earnings results. There must be something they have did relating to supply constraints as well. Its good for them to start selling in the future for sure.