1. The toilet horn may sit inside of the closet flange bowl. A rope of modeling clay could be used to dry-fit the toilet and measure for clearance. Wax does not need thickness, but you don't want the toilet to rock on the flange. Lifting the toilet a tad with shims could accommodate if it's close. Always have shims in place before dropping the toilet onto wax.
2. I think some toilet horns don't get as close as 3/8 to the level of the surface, but I may be mistaken. Try using a straight edge and measure carefully. Or maybe measure a different toilet. If that is an older toilet, a newer one might be a nice upgrade, plus accommodate the thick ring.
3. Could it be that the existing flange is a Push-Tite 886-GP? If so, you could replace that with a 887-GPM, and have a much thinner stainless steel ring.
https://www.siouxchief.com/products/drainage/residential/closet-flanges/pushtite
4. If that is a PVC flange into a donut or some other means of connection, how about you sand off the upper 1/4 inch of PVC with a belt sander. Then put a repair ring over that to give the strength you need to hold down the toilet. Those come in different styles, including those with ears that let you hold that down with screws outside of the PVC ring. I am not a plumber, and this last one is not a normal thing to do.
5. A Toto Unifit toilet might work. The Soiree or Vespin II for example. If interested in that, there is more info. Those are not low-priced toilets.
