Don't buy a budget semi-auto. If you can't afford a good semi, buy 2-3 pumps for the same money.
Gas operated semi-autos with shorter barrels tend to be less reliable. Especially with anything other than magnum shells. Not less reliable, but unreliable. In other words, shoot only magnum, or heavy loads in them.
Cutting a barrel on a gas operated semi-auto really screws up things. They are tuned for a specific barrel length for the gas pressure to be right to operate the action. Don't buy a SX4 and cut the barrel. Cutting a pump isn't a big deal.
My brother shoots lefty. He had to choose between left-handed guns he didn't like; or learn to shoot right-handed guns he did like. He chose the latter. All of his guns; AR's, bolt rifles, and shotguns are right-handed, and he has never had an issue using them. If you choose a model with the safety button behind the trigger it is easy enough to use it from either side.
I have an older Benelli M-1 with a 24" barrel. That is a little longer than most use for HD, but it is short enough. And long enough to be reliable with any load. Being inertia operated, the barrel length is less of a concern too. For me it is a good dual purpose hunting/HD gun that doesn't look "tactical".
Street price on the newer M2 is about $1200. I just ordered one in 20 gauge with the 24" barrel and my price was $1200 OTD. It's in on a boat in the Atlantic right now and I should have it in a few weeks.
https://www.benelliusa.com/shotguns/m2-field-shotguns
I'm not sure if they have that in lefty or not.
Another option was discussed over at "The High Road" recently. This is right-handed only, but street price is supposed to be around $1000.
https://www.thehighroad.org/index.php?threads/new-release-beretta-a300-tactical.914906/
That's a proven design and a lot of gun for $1000. It has some tactical features you may not want and is right-handed only. It will also probably only function with heavy loads with that barrel length. But if I were looking for a semi-auto HD shotgun, I'd look hard at this one. Even if I shot lefty.