How about, "It's cold outside, put on the coat."
By pealim site, it can mean put on. https://www.pealim.com/dict/973-lilbosh/
I think "put on" would need a different verb.
Why is it לְבַש for לובש, but for שואל its שַאַלי? Do people also say שְאַל?
It's feminine vs. masculine forms (שאלי is feminine, שאל masculine; לבשי is feminine, לבש masculine).
In correct English, we would say "wear your coat" and not "wear the coat". DL should accept "wear your coat" as correct, even though the Hebrew sentence does not include "שלך".
"Wear the coat" would be used if the other person has been wondering which garment to wear. "Should I wear the sweater or the coat?" Answer: "It's cold outside. Wear the coat."
I agree - correct English would either be "wear A coat" or " wear YOUR coat"
In hebrew this sounds more like "your coat" not "the coat", because the verb is conjugated to YOU.