This decision will drive how much light power the LED will need.
The project that you show is 8mm film (Regular and Super 8). Typically the sphere required is not large if it is for 8mm only.
High CRI is somewhat the opposite of discrete RGB LEDs. It may be best to look at the White LED datasheet spectrum profile.
These are the White LEDs I am using (5000K), with a custom designed PCB. The LED subject and quality is a recurring theme in the forum. If the interest (and patience) is there, this may be an opportunity to collaborative design the best form factor for a PCB.
The PCB at the bottom is what I have been using with the LEDs above, first in a small oversided ping-pong ball, and now three in a 90mm metal sphere.
The SMD LED retail cost is about US 35 cents, and PCBs are not that much.
Note that these LEDs have a particularly large Vf of 9V, so typically it will require a 12V power supply (or more if connected in series). In the large sphere above I use a 32V power supply.
PS. As you can see from Manuel results, you can have good results without the complication of building the LED/PCB, so please take the above as an illustration, not a recommendation.