Great question, this is always a concern for us when booking sites. Generally tents sites are the best and we like to be away from the big rigs and generators. So I will always choose a tent site when we are staying at an actual campground and not boondocking. To this date, knock on wood, we have had no problems. If it's a tents site that you pull your car up to, you should be fine. As someone else mentioned make sure it's not a walk in site. My worry is that the personality of campground hosts can vary greatly. Most are very chill and relaxed but some are militant about rules and may make a case that you can't be there. This has not happened to us but if it does we have an argument prepared about how our campinn is just a hard sided tent on wheels..........
I think the key here is you called ahead. Do your homework ahead of time and am sure you will be welcomed - rather than just doing it.When in campgrounds I always take the tent sites and have never had a problem. Recently we camped at the Belknap campground in the southern Sierras (an outstanding place, BTW) where bold print specifies "Trailers are not permitted!" due to the tight curves in there. I called the forest service office for the area and described my trailer and was told that I wouldn't likely have any problem and to go ahead. He was right.
Take advantage of that dinky trailer!