I would not recommend anything that involves engaging your core until you have worked out the root of the problem, two years ago I was suffering from whooping cough, I was stupid enough to carry on training (powerlifting) while I was Ill, anyway..
I coughed / strained during a deadlift and felt a strange sensation In the right side of my abdomen. When I woke up the next day I had to go to hospital as I could hardly move without screaming in agony. 3 days in hospital and I eventually recovered, but it still took a long time for the right abdomen pain to go away.
Since then I occasionally found I had a dull ache in my testicles when I tried my 1rm (bench/deadlift/squat)
It only happened occasionally but I carried on training...
Two years later now and i can't lay down on my back without feeling discomfort in my left nut...
Turns out I have a misaligned pelvis , FAI/ hip impingement , athletic pubalgia/ a possible inguinial hernia
^ when it comes to testicular pain with heavy lifting , hernias and pelvic floor dysfunctions are usually the main culprit
IMO It would be wise to see an orthopedist to get a better idea of what's going on.
Take it from me, rest and find out whats going on,
You do not want to make this any worse.