I am sorry but No.So generally, when do we need to put という before くらい? It seems in many cases we use just くらい after a verb.
The condition that triggers という before くらい is a more structural one than "direct relationship." Take a look at the first sentence. I will quote part of it below (to thetazuo, 簡体字 redacted):In sentence 2 you can see direct relationship between the two:一歩も歩けない and 疲れていた while in sentence 1 there is no direct relationship between 1日が始まらない and コーヒーが好き.