I was planning to do a full game run but is wondering if a timer is needed to be shown on screen

No. When I do any % runs, I calculate the time after I complete the run. Because I don't have a capture card, I record on my phone and compare the start and end times of the video. Not sure how that would work if you have a capture card

A timer isn't required for full game runs, regardless if a timer is there or not the mods retime each run anyways