The As-3900, if unmodified, would have presented a reasonable load to the CB transmitter but won't radiate much on CB frequencies. If "gutted", and a coax connection made to the feed point inside the base, the antenna now looks like a dipole for a much higher frequency than CB. (I am remembering mid-30-ish MHz.) That high resonance is the design of the antenna as a dipole with feedpoint near the middle, where the two antenna sections screw together. Half the dipole is the length of the short top section of the antenna, hence the high frequency. That length is nowhere near CB frequency. There is now no match to the antenna on CB. It might still work ok on receive but the transmitter won't like it.
What to do? I would get an automatic antenna tuner, like the LDG Z-11 PRO. These show up used but aren't too expensive new. Connect the radio to the LDG radio port, and the LDG antenna port to the modified AS3900. The tuner will match the CB to the antenna so the CB will like what it sees. Radiation from the antenna will not be optimum (like a true 102" whip) but I'd guess the signal output will far exceed what you would get out of the unmodified AS-3900.
Search on SS for AS-3900 and AS3900 because there is a lot of info already here.
On your grounding question, ground the antenna base's housing, and ground the coax shield to the housing. This won't ground the "antenna" because it is a dipole and a dipole does not need grounded.
