I just started climbing, should I downsize beginner climbing shoes?

I wouldn't. Downsizing isn't really meant to be a beginner thing, it's really not beginner friendly. The whole point of downsizing is to have better control and performance for the more difficult and advanced routes, beginners don't climb on advance routes that require high performance shoes or downsizing.

I would hold off on downsizing and only do it when you get to the higher grades and are more knowledgeable in climbing and climbing shoes, it's a whole world out there!