Plywood subfloor won't screw down in a trailer home
When you say "bite down" do you mean:


It actually goes through the layer beneath the newly laid 1/4" plywood and simply spins as though it's stripped and will not counter sink, or
It will not even sink far enough for the head of the screw to even touch the new layer of plywood?


If it's option 1, are you using a wood screw or machine screw? A machine screw (self tapping or otherwise) would not be a good option for use in this scenario, or most scenarios when the underlying material you're fastening to is wood. The threads are too fine and tend to strip the holes easily. You should be using a wood screw, which has fewer threads per inch (TPI).

If it's option 2, I would agree with @JPhi1618. Consider using a shorter screw so you are not piercing the underside of the floor. This could potentially allow moister to wick in from the underside and rot out the exposed ends of the screws and eventually the subflooring. At minimum, the driving depth of a fastener should be no less than double the material you are fastening. In this case, you are fastening 1/4" plywood, and the math says 1/4" x 2 = 1/2". Therefore, your screw should be at least 1/2" long.

I am guessing that what you mean by a "stick down tile", you are referring to a peel and stick flooring? Similar to  found here Home Depot link to the example picture

Personally, I would be hesitant to use a concrete backer, simply because of the amount of dust that sits on top of the backer regardless of how many times you sweep or vacuum, but that's just personal choice. Regardless, I would definitely suggest priming the surface to help ensure a dust-free, clean surface so the tiles will have a better stick-to-it-ness (I just made that word up) and is less likely to have corners curl up prematurely or bubbles appear from what might seem "out of the blue". Best practice, for sure, is to follow the manufacturer's suggestions on underlayment and surface prep.

Hope this helps!