I am no expert, but here is what has worked for me.
Backstory: I am naturally kind of fat. Pre-SS I was 225# and fat. After several months of SS and then 5/3/1 I was up to 240# and much stronger but still fat. I am now about 220# and am gaining strength and slowly surpassing where I was at 240# strenghtwise.
What has allowed me to progress while eating clean is a reduction in training tonnage. The biggest thing I found to help reduce tonnage was doing ramping sets instead of sets across. I do 5 ramping sets of 5 up to one top set of squats and bench/press. I do 5 ramping sets of 3 up to one top set of DL/power clean. The first set of 5 is generally 50% of the top set and I work up in even increments from there.
The next thing that helped was reducing training sessions to 2 days a week and arranging the workload in a Heavy, Medium set up. See Andy Baker's explanation at bakerstrengthcoaching.com for heavy, light, medium training.
Day 1 (Heavy): Squat 5x5, Bench 5x5, DL 5x3 - All sets ramping to one top set
Day 2 (Med.): Squat (90-95% of Heavy) 5x5, Press 5x5, Clean 5x3 - All sets ramping to one top set
Increase top set of Squat/DL every week by 5#, increase other lifts by 2.5#. With this program you can do conditioning pretty much any day expect for the day before your Heavy training.
I know the post was long winded, it took me several months of f-ing around before I found something that worked. Hopefully this will give you some ideas to save you some time.