I would say the pickup cover isn't required at all. If anything, it will introduce hum like it does on all of my P basses. It's not necessary for the Jamerson tone unless OP needs to anchor their hand on it in order to pluck in that zone, and Jamerson didn't rest his hand on the cover from what I can see.
I naturally pluck right between the pickups and the fretboard (due to narrow shoulders and long ass forearms) with or without the cover and feel no difference. I anchor my thumb on the pickguard
Jamerson seemed to pluck closer to the cover than the fretboard, and that makes an audible change from my plucking spot, though it's subtle.
The bridge cover is more of a requirement, however, because in order to achieve short sustain on the low strings and longer sustain on the high strings, the foam block should be placed above the strings, not below like everydone does. Something like a Nordymute would also work without the need for the ashtray. If the only option is having the foam under the strings, then cutting the block at a slant will achieve similar results.
If you use the Jamerson Labella set, the E saddle must be adjusted way back to intonate the string, so make sure to angle the foam accordingly, otherwise the E string will be sharp. This is really only possible if you glue the foam to the bridge ashtray.
This is also nuance that is largely unimportant towards the overall tone but hey, everything matters to an extent