Causing someone else to sin is a sin. If you ask a doctor to lie for you its probably worse than lying yourself because you are possibly condemning someone else.It is bearing false witness. That being stated, the doctor entering his name into the database it technically the one lying. Your friend could always just answer, "my name is in the database", and never say anything more. I wouldn't use a non-local doctor though, especially one taking money from people across the country to do it. It would look suspicious to any investigator. The issue is future booster shots. What if those become mandatory? Is your friend going to be able to sustain this? Personally, I won't even wear a mask because I think that's a lie, so there's no way I'm lying about getting injected. I'm not going to compromise with evil.
This reasoning is kinda like all the gymnastics the Jews do to get around the laws. It's Phariseeism. Encouraging others to lie for you, so you can keep your conscience clean is a sin, let's be clear about that.