I heard a “Rebelrocker” bass idential to yours in a local shop. Your comparison was reasonable, but modding the fake is not a fair representation. Potentially the 4004 could be a cheaper model but then it wouldn’t be a 4004. The only “budget” model I think should exist is a 2000/2020/3000 style. A lower quality/cheaper Rickenbacker 4003 is not needed in the market and I venture to say few would buy one.
Rickenbacker runs their particular type of race for a reason.
Maybe they could license a company that never makes them… if only for legal recourse. Rickenbacker licensing these is in my opinion not a good idea.
