What would you suggest? I might prefer to stay within the Pro-ject line as I could reuse the acrylic platter (if one is not provided with the recommended turntable).
No offense, but not the greatest reasoning. You've already got some good advice, in terms of baby steps are not the way to go. Your goal should be to save up for one that will be not a little but a whole lot better, and last you for years.
Personally, I think the best approach from here on out is to consider table and arm separately. For several reasons. It is usually hard to save up. For whatever amount you can manage, it will get you a lot more quality spent on just the arm, or just the table, than a package.
Also those who make arms to be sold separately, they really have to make a good arm because people are going to compare. That being the whole point. Whereas those like VPI making packages, pretty much nobody ever takes one of those arms off and that is how they get away with BS like their twisted wire side bias. Not to pick on VPI. Same goes for everyone else selling packages. I just like slamming the dumbest side bias in the business.
Finally, buying them separately forces you to learn a whole lot more about the value of each part, how it works, and how it sounds. Vinyl isn't like digital. You don't just push a button. The sound quality you get is as much to do with your skill level as anything else. Skills cost no money, but pay off big time. Invaluable difference.
But you deserve an answer for your $1k. The best one I know is it makes no sense to upgrade to a new component before extracting full performance from what you have. The more constrained the budget the more true this is. So put that table on some Townshend Pods, or a Platform, whichever you prefer and can make work. This will almost certainly be a greater improvement than the same money spent on a new table.
If you want to go cheaper, and test the waters, get a set of Nobsound springs. Only $30. Then when you hear how good this works upgrade to Townshend and move the Nobsound down the chain to your amp or phono stage or whatever.
The more experience you get with how music is supposed to sound the easier it will be to ditch that expander DSP stuff that is messing up all the good vibes coming off your vinyl. Selling those will free up funds for that new table and arm.