"My son lost my tennis racket."
Translation:우리 아들은 제 테니스 채를 잃어버렸어요.
"우리 테니스 채(를)" would be "our tennis racket". The English sentence, however, specifies "my" tennis racket. I don't think that the exception of 우리 sometimes meaning "my" in Korean would apply when you're talking about things like tennis rackets. But I could be wrong or nitpicking here.
I agree that 제 should be accepted at the beginning when saying "my son", but it would still be more natural to say 우리 아들은.
You do not really need to be "consistent" when translating the word "my" into Korean. Because the fact remains that sometimes, 제 is more appropriate, and at other times, 우리 is more appropriate. And in this case, "우리 아들은" and "제 테니스 채를 " are the most appropriate translations for "my".
Very interesting the use of possessive adjectives in Korean.
The focus is not on consistency of their use in a sentence but rather on the individual nouns they describe.
So in general regardless if those nouns appear in the same sentence or not:
우리 (collective ''my') is used with collective nouns s.a. country; school; loved ones etc.
제, 내 (personal 'my') with nouns of personal nature s.a. razor; wallet; racket; etc.
But like you said there is a grey area where either types of adjectives may be applicable depending on the perception of the speaker; nouns s.a. house; friends; etc.