The integrator will change the size of the beam but not necessarily the angle. For example the beam at the entrance may be 50mm and the integrator may reduce it to 30mm (just a geustimate

) but the angles will remain roughly the same.
What I mean by small lens, is just a plain lens
not a reflector/lens combination for the high power LED. Your LEDs already have a 15° beam angle, by using a plain lens (PCX), the 15° beam could be made to be closer to 0° which is what the integrator is expecting.
There is another cause of light loss by not using collimated light. The integrator and PCE will both be working against you. Take a look at
this description I posted recently. If the light isn’t collimated before entering the integrator then it won’t converge down to pass through the PCE and allot of light will be lost.
As to where to find small lenses, maybe surplus shed will have something. Or perhaps cannibalise some disposable cameras. What you will be looking for is a lens with a FL that is not too long as it won’t fit into the space you have. For example if you have 30mm between the integrator and the LEDs then you’ll need a lens that has about a 25-30mm FL. The diameters should be large enough to match the number of LEDs you are using and still fit into the entrance. Plastic would be preferable as you could cut them to size.
DJ
Edit:
Small credit card sized fresnels are another possible lens. As long as the FL was correct.