Should you be sore after pratice or playing can be answered in one word.
The lips may be tired for a few minutes or the mouth corners for 15 minutes or so , but anything more that this is a sign that you have caused some damage. As for being sore until after the next days warmup. Well that is way off base.
If you feel sore or tired the next day then you bruised them and there is some swelling present. A competent teacher would never advise a student to play until that level of soreness was reached.
Also that kind of soreness can only come from excessive mouthpiece pressure. What is normal and acceptable mouthpiece pressure for compressed lips (cushioned) is too much for flattened chops. You would be amazed at how many players set their chops and then unset (thin the lips and spread the embouchure) as they take a breath. This is the case in 80-85% of the cases that you described. I have not seen you play therefore I can not say 100%.
No amount of lip pushups will cure this. Isometrics were only taught by Stevens to people who had a viable embouchure. For people who need some embouchure help (fine tuning) the embouchure must be fixed before the gross muscle building program begins. Remember holding a pencil between your lips is not the same as playing. One is for gross muscle building while the other requires a great deal of finesse training.
Finally I would say that if you bruise your lips on a daily basis then the setting is spread. You can only play when mouthpiece pressure flattens them enough so that they are touching.
A visualizer is only to check where the center of your buzz is. It is still possible and likely to push against spread lips to force out a buzz. Also I’ve seen many players start a session right and distort the lips as they breathe. So a 20 second buzz with no breath included while using a visualizer would again be useless.
I would advise you to learn to buzz just your lips. Mendez made all new students buzz their lips 30 days. After 30 days they got a mouthpiece. After another 30 days they played the trumpet. The backpressure from the trumpet is a crutch. As is the mouthpiece. Plus buzzing the lips uses all of the playing muscles at once unlike other isometrics. Buzzing the lips uses the entire length of the lips. The mouthpiece divides it and prevents over half from buzzing. Set your lips close together, breathe through your nose (only for this example) and buzz them. After a couple of minutes of this sneak the horn up into playing position while you are buzzing. You should have a very full sound with no air. You will also need less mouthpiece pressure.
I like to advise some buzzing 15 minutes any day that you don’t practice. While it is true that hours of buzzing will stiffen the lips 15 minutes will not. BTW hours of isometrics will stiffen them also but with fewer tone building benefits.