You have a common problem my friend, champagne taste and a beer budget. Simply put, decent quality side by sides cost more to make and sell in much smaller volumes than over/unders. It also is much more difficult and expensive to make a lightweight version than a heavier one. Actions must be more petite while still being robust which requires careful design, choice of materials and precise assembly. To make a shotgun lighter while staying in balance the barrels need to be lighter. This normally means thinner which requires stronger (more expensive) material which in turn requires more careful attention to tolerances during manufacture. Wood weight must be appropriate to the intended balance and the whole gun is very difficult to balance while keeping weight down, the lighter the gun the more difficult this becomes. Many of these tasks need to have the personal touch of experienced human hands. Yes, you can buy a clumsy overweight small bore side by side that feels like a 2x4, swings like a fence post, shoots both barrels to different points of impact and quickly loosens up. There is always a market for inexpensive guns like this and they sell quite well because many buyers don't know the difference, won't shoot it more than a box or two per year anyway, and they let their pocketbook rule rather than the desire for quality.
As mentioned, on the used market the Ithaca/SKB meets your specifications, some Spanish guns from well known makers, some older lesser known Italian makers, an English 20 if you can find one, could be just what your heart desires. The Browning BSS and the Winchester 23 are fine guns but heavy and well over your budget. Sorry, you need to start saving because most of these will come in well above your $1000 limit ( beer budget)
So what's available new? Many Turkish guns such as the CZ, Huglu Churchill and others, while being over your weight limit as a 20 are very trim and lightweight as a 28. And don't whine about the price of ammo, maybe a difference of a cup of coffee between 20 and 28. Sub $3000 the FAIR ISIDE can be had in less ornate trim for well under $3000, very light and well balanced, reliable and can be had with different barrel lengths, fixed or removable chokes, single or double triggers. Start saving now, patience is a virtue.