The problem you will encounter is making sure the fiber optic cable (FOC) is directly above the LED. As the "just squish the LEDs as close as possible" method means that you will need to measure the distance between all the LEDs and then drill EXACTLY where the center of each LED occurs in the rigid panel. This is more difficult than you may think. An alternative method would be to put two pieces of the rigid material together (one on top of the other) and then drill holes using a drill press straight down through the two pieces. These holes need to be just big enough on the one to put the FOC through it and the other to be just big enough for the LED dome or chip (depending on the style of LED you use) to protrude or be centered in the hole. Put the two pieces back together and you are more assured that the FOCs are now more centered. Additionally you will lose light if the LED "box" is open. It would be better to try to enclose it and make sure that the LEDs are as close to the FOCs as you can. That being said, LEDs still produce heat that will need to be dissipated. Not a lot of heat, granted, but it can reduce the lifespan of the LED. Modify the apparatus as needed (spacers between the two rigid material pieces, ensure air flow through the box, etc).
The use of lenses to focus the light will work, but require a way to make sure that each LED is pointed correctly and not off a bit, producing a mess. Also, it may be easier to just bundle the fibers into a round cable instead of a square one and then take it directly to the white acrylic lighting dome. Are you really wanting to focus the light, as each thing the light has to pass through will diminish the amount of light passed? You will encounter light losses with each transmission, so keeping it simple will increase the amount of light you have at the end.
Good luck.
For disclosure purposes: I work with lasers and FOCs in the instrument I run at work. Granted, you aren't using lasers and you don't need it to be as exact as I need it to be, but the manufacturer uses a three dimensional focusing method to make sure the light source is aligned to the cable. If it isn't, you get a much more diffuse and much less powerful beam; for my purposes, it becomes basically useless, but for your's you may not have to go to such extremes. Lasers also produce heat. The amount of heat it produces will depend on how the laser light is generated; the instrument I run uses Class IIIb air-cooled lasers at about 50 mW (or greater as stated on the fact sheet from the manufacturer) but I would have to talk to the service engineer as to what maximum he would allow from the lasers. This machine has some big and loud fans on it just to keep the lasers cool and also to keep the fluidic lines from getting too hot. As the heat increases, the viscosity of the fluids used in the method changes, making the job of the instrument that much more difficult. Also, the instrument uses focusing prisms to help collate the light and send it to to required spot.