I cross my straps on this Serpent Express, mainly due to the slippery floor. I never did in my enclosed trailer, but I did have the floor painted with a non skid type paint. Read this from Hagerty, it says you can or can't, sounds like personal preference. I think the idea is to keep the rear of the car from sliding around, but, the front would be as important. The main thing is to use the proper rated straps for CARS, not motorcycles. Breaking strength is different than Tensile strength and so on....do some homework and use RATCHETING straps, not pull tight straps.
As Ian suggested, front wheel chocks are helpful, however, the drawback is being able to properly tighten the front straps. I have mine marked on the trailer, I put the wheel chocks down for loading, then tie down. Once all the straps are tight, I put the front chocks back down. Your idea/thought of using fixed straps has some real merit and something I may try.
When loading your car, either have the trailer connected to the tow vehicle OR have jack stands/adjustable jacks on the rear of the trailer and chock the tires, front/rear. As also mentioned, make sure you have enough clearance for the front of the car to not only clear the rear gate, but clear the break over point where the gate meets the trailer (namely oil pans). A few ways you can adjust for this, you can jack up the front of the trailer lessening the angle of the gate to the ground. Park on a hill with the trailer/tow vehicle pointing up the hill, or buy the SpeedRamps. Up to you.
https://www.hagerty.com/assets/PDF/TraileringPamphlet.pdf