You want a quality red dot for a budget price? Good luck with that. It's just the way it is. Maybe find one of quality that's been stolen but there are laws against buying stolen goods. I have a really hard time suggesting something that might get someone killed if they use it for HD and it should fail. So I'll start with what I'm confidant will work 100% and go down in price from there while still remaining reasonably confidant that it'll work 100%.
OK, bear with me for a bit. I shoot competitively and that doesn't involve life in the balance, it's just a game. The firearms of serious competitors don't wear inexpensive sights. Why? Because they need to work every time, and it's merely a game. What I shoot is timed and there is no such thing as stopping the clock to change a battery or fix things once the clock starts. HD is much the same. You aren't going to be able to tell the bad guy to hold on while you fix things so that you can ventilate him later. I do have a shooting buddy who uses an older sight with lousy battery life. I've seen his sight dim down in use so that it was unusable. In a shooting game it might be OK, in HD, not so much. It can get you killed.
You're not the first to want quality but not want to pay for it. If you truly are in need for HD, I have to ask what your life and families life is worth? If you're just setting something up and know that the odds of ever needing it are really close to zero then anything will work, even if it fails. But if I was taking HD seriously I wouldn't look for quality at a price where quality (meaning reliability) isn't typically found.
It's out of your stated price range, but Holosun has reasonably priced quality sights with auto brightness adjust and shake awake function that will do what you want. Anything else with the same quality would be more costly. If you like your laser, and I know nothing about yours, what would be the max' distance of employment? Maybe the laser alone is enough? Typical HD range is quite short so it might be sufficient. If you mean HD outside the home then distance can stretch, and it's also very hard to justify as HD. Shooting someone outdoors at 50 yards and claiming HD isn't something I'd ever want to try.
I have lots of Holosun sights and they all work fine, are reliable, and I have many thousands of rounds fired under them. I save a small amount of $ by shopping around to get what I want for the lowest price I can find. It takes a bit of time doing a search of the 'net. I shop for specific model(s) that sacrifice nothing that I want for features and I never take just anything that I find that saves me $. Sometimes I might not get immediate gratification, but I'mm in no hurry most times. Maybe you can save $ over time and get something that will be of higher quality?
Features that I want as a minimum, are sleep/shake awake, and solar (PV) cells, both of which lengthen battery life. Look for a long battery life certainly 20,000 hours or better and still change batteries every year for an HD gun. Brightness auto-adjust is also mandatory but know the weakness of it. If the sight is in dim light and one is shooting into brightness the auto adjust will yield a too dim reticle. All Holosuns have a manual brightness adjustment. On a rifle I also want an enclosed tube and not an open "projection" type of sight, but for indoors and not out in the elements that's not a big requirement. I like a QD mount so that if I must I can put a new already "sighted in" spare sight into place and continue the match, but not having that will save you a few $ and isn't needed for HD. Not getting an anti-reflection filter also isn't needed and will save a few $. Holosun models can be had with full military features, but again, you don't need them and the more that's added means higher cost. A stripped down Holosun with just the bare features you need might be affordable.
SIG also has sights that I wouldn't feel horrible about suggesting, but the ones I see for near $100 (Romeo5 ?) just aren't what I would consider full featured. But if you remember to change the battery every year they'd probably do what you want. I know folks whose opinion I respect who like them but I have no experience with them myself.
Primary Arms also had (has?) not too costly red dots, one that works fine for me. Again, not what I would consider full featured, but they work fine, are bright enough for full bright sun with full snow cover, and have decent battery life. I have one that has been in use for years. It's also not on a firearm that I would think about using for HD not for any reason of reliability or any such thing. I used it for years on my competition sub2k and it's still on it and still working fine. I think they called it the advanced red dot or some such. Maybe they still make it? I seem to remember it being near $100 on sale, certainly under $200 at the time since like you, I was on a budget. PA also stands behind their products. The model I have has an easy to adjust manual brightness wheel and goes very dim, in fact it has NV settings which you won't need.
Good luck with your search. If you find something and are unsure maybe ask here? Obviously you'd want to link to it.