I happen to have recently added the RTX 3060 12GB onto a i5 3570 (so a fairly similar setup to yours) and it can run something like Baldur's Gate 3 in 4K all settings on. I can also tell you that prior to that, I had an AMD RX580 8GB in that system and it ran Baldur's Gate 3 with most settings on at 1080P and ran up to around 45FPS depending on where you were.
Given the RX580 is roughly analogous to the 1650, I would avoid that card (1650) if possible. So I would suggest anything from the previous NVidia generation (2060 or onwards) and their AMD equivalent cards for selection choice (depending on pricing and budget you can afford). But don't get the latest cards, as you'll likely be missing out on performance that the CPU can't drive the GPU at (and also need more power to run it from the PSU). For example, that AMD RX580 8GB? Previously in an older X79 (4930k at 4.2Ghz) it couldn't run Baldur's Gate 3 above 45FPS at 1080P in the best of situations (usually 25 to 30 FPS instead). However, once it got into a Ryzen 9 7950X system, it was able to drive it closer to 60FPS. But all of this also required more PCIe 8 Pin connections from the PSU, so you'll need to factor that in your budget. The 3060 I got was because it only needs 1 PCIe 8 Pin power connector, so fairly easy on the system. However, in your case of the 350W PSU, you'll likely need to swap in that 600W one you have stashed away (assuming it has a PCIe 8 Pin available).
