I can't comment on the conversion re the glass doors but if your older Billy cabinets have pre drilled holes, then it should be easy to order the doors for them and install them with the included hardware.
Be warned that the colors of the wood grain have changed. My two older Billy Oxberg cabinets were a medium brown wood grain effect, which are not available. Instead, for my third cabinet I chose a dark grained wood as the only suitable alternative.
The newer cabinets come short one shelve compared to the older version. So, you will have to order at least one more shelf (or more if the distance between shelves is reduced to hold more figures. The space between the moveable shelves is predicated on the the permanent shelf. The later is part of the bookcase structure.
The distance between my shelves varies from 8 1/2 " at the top to 14" at the bottom (for tall items like ships etc.. I believe my shelf spacings are roughly 8.5". 9", 9.5", 10", 10.5, 11" and 14". However, the bottom shelves could be rearranged to add additional shelves below the main/permanent shelf. Note I store 28mm figures and the distance I use allows me to easily remove and insert figures with ease.
I used LED adhesive tape lighting of Natural white at 60 lumens per meter in 8ft rolls. BTW, I would not recommend this strip lighting for glass shelves as they might not stick well and the reverse side of the tape would look unsightly through the glass. Of course the number of rolls depends on the number of shelves you have. Also, the tape can be easily cut with scissor to fit your needs.
Luckily I have found that three Billy Oxberg display cases side by side allowed me to span one 8 ft roll of the LED tape lights for one level of shelving for all three cabinets. The tape was affixed to the underside roughly at the half width point of the shelf starting from right (for me perhaps from the left for you depending on your configuration needs) along the shelf to the left toward the join of the two cases. I previously drilled a large enough hole through the two cases to permit the tape to fit through. I repeated this procedure for the next cases, which completed one complete shelf run for the three cases.
BTW you should leave the protected tape covering (the sticky side) on until you have pre-fitted the full length of tape through all the cabinets with connecting wires out of the left side of the last case. After pre-fitting gradually peel and stick the tape to the mid line of the shelves.
I repeated this process for all of the remaining shelves. I then connected all the wires for all my shelves to dimmable high wattage power supply, which was affixed to the left side of the case. I recommend choosing the highest transformer wattage possible to allow for possible expansion and mitigate under powering your LEDs.
I hope my method of installing these LED tapes helps you with your project.