This is an interesting concept I have been thinking about. I have been doing LED conversions on mostly channel letters & a couple of odd shaped signs, all filled initially with neon tubes, I had been using Sloan LEDs which in my opinion are very expensive. On a channel letter conversion, you have to send a full size vector mock up of the sign to Sloan & they will configure the amount of modules needed & how many transformers to hook up.
The last time I called for LED modules to my supplier (NGlantz & Sons) They offered me some LEDs made in China. I had to send the layout to the distributor in California & they sent me back the recommended LED layout. The job was 36 inch Honda letters, After ripping out the neon & installing the LEDs the customer was ecstatic about how well they light. The LEDs were as bright and as easy to use as any & were 1/3 the price!
Which brings me back to your post, I think these may be a good replacement for the
Florescent bulbs in the cabinets, if the sign would be single sided you could place them on the backer, if it is 2 sided you would need center strips or a center divider panel & place 2 times the amount of LEDs, which would be somewhat costly but to the educated customer, much less hassle than changing bulbs & or ballasts every couple of years & probably brighter & more energy efficient. If you build your own cabinets from a knock down kit you could easily add LEDs instead of traditional bulbs.