Unless you are showing your sheep, I'm not sure why you would be concerned about hay belly. Sheep need that long-stem forage for their rumens to operate well.
Club lamb people have all sorts of tricks for keeping their lambs sleek and trim, but I have never raised club lambs.
I know when we show our wool breeds, we generally withhold hay the morning of the show to reduce the extra belly during the show, but besides that, they are allowed to be sheep and keep a nice full rumen. On my farm, grain is used as a protein and energy supplement on top of whatever grass or hay they get. It is used to support growth and development and is never the majority of their diet.
I started the lambs out on a 17% textured feed that was way too expensive for my taste. They are now on a blend of 16% pellets with cracked corn. I top dress with soybean meal to bring the protein level back up to 16-17% protein. I like a pellet that is medicated for coccidia, especially in the 1st couple months of their lives.
The ewes don't get any grain except going into breeding season (flushing) or shortly before and during lambing and during lactation.
What do you have for sheep? How and what you feed really depends on your breed and purpose for having sheep. My flock is mostly purebred, registered Border Leicesters, but I also have some registered Shetlands and a handful of crossbred ewes for meat lambs.
The Shetlands don't need nearly as much supplementation with grain as the BLs and crosses, but because they are housed together, they get the same percentage of grain as the bigger sheep.
I feed my lambs 1.5-2% of their weight in grain daily. I run them over the scales every couple weeks and adjust their feed accordingly. They have 25/7 access to either grass or hay.
Unless they have bloat, what's a bit of a big belly between friends? My lambs are all grass-fattened these days and do tend to have fairly large bellies due to all the methane they are generating. Avoidance of too much/too soon with grain feeding is very important in order to avoid bloat, though, when feeding grain/