I don't have many answers for you but a question. My agency's employee handbook says that part-time is an option and spells out how pay and benefits work (though manager approval is required and in practice I've only heard it used for parents of young children and an older person on dialysis). I'm not ready for part-time yet, but I've never been able to find out what forms to complete to ask for it. If you find out the form(s) involved, I'd appreciate you letting me.
You said you want one day off every pay period. My employee handbook says part-time is 16-32 hours/ week, so one day off every other week would not be an option. I don't know if that's agency policy or government-wide.
Re: health insurance, you can still get it part-time. At full time, gov pays roughly 2/3 and you pay 1/3. If you work 80% of full time, the portion gov pays is 2/3 x 80%.
The amazing thing about part-time is is that part-time years count as full years for MRA and an unreduced pension, i.e. if you worked full-time for 32 years, you could retire at age 57 with a pension 32% of high-3. If you worked full-time for 10 years followed by part-time for 22 years, you'd still qualify for an unreduced pension age 57 of 21% [10 + (22 x .5) x 1%]. Your high-3 is considered what your full-time salary would be and years of service for part-time is number of years x % of full-time. Seems like an amazing deal if you can get it.