Newly commissioning officers should purchase the new uniform.
Regarding the Mess uniform. It is customary for Field Grade Officers to have the uniform - considered a right of passage to mark the transition from directly leading Soldiers to leading through subordinate leaders. But, it is not mandatory.
However, plenty of company grade officers own mess uniforms. I never met an officer stationed in Korea as a LT who did not own a Mess uniform for example.
If the LT is purchasing the greens, buying him/her the mess would be a fantastic gift, and undoubtedly much appreciated IMO.