For this reason I have spreadsheet-ified much of my organisation for life, so I can be active on my laptop even if I have no work.
I have a massive Christmas spreadsheet, household organisation spreadsheet, meal planning, etc, you get the picture. Is there anything you can do like that? You say you're gaining weight, can you start an excel weight loss/fitness tracker? They have some amazing templates!
If you were actually asking me, I love audio so I would bring my headphones with my phone and have a list of audio books I wanted to get through, with some podcasts and music lined up for breaks. That way you can keep your PC set on some "work" and there's nothing to be caught out over. Say you're doing data deletion to keep you in GDPR or something.
I'd find it hard to believe that anybody in a quiet and isolated office with a light workload day would be begrudge by a manager for having some headphones on while working.
If you're genuinely bored and want some work to do, you don't have to out yourself and say you absolutely fuck all to do, just say you have some additional capacity today as you will be finished your current tasks by x time, and see if you can be given something.