I went through this exact issue when upgrading my editing rig for 4K Sony footage. Premiere Pro heavily relies on Nvidia CUDA cores, so you definitely want to stick with Nvidia for this. Here is what you need to do:

• Buy the MSI GeForce RTX 4060 Ti Ventus 3X 16GB OC. The 16GB of VRAM is crucial because Premiere eats up memory when handling 4K timelines and color grading. It fits right in your $450 budget.
• Enable hardware acceleration. Go to Project Settings, then General, and set the Renderer to Mercury Playback Engine GPU Acceleration (CUDA).
• Install the Nvidia Studio Driver instead of the Game Ready driver. It is way more stable for editing apps. This setup should stop the timeline lag immediately.

I bought the Gigabyte GeForce RTX 4060 Ti Eagle OC 8GB for my travel vlogs last year and I am so satisfied with it. My 4K Sony footage works well now with no complaints. A couple quick tips:

• Grab the 8GB version to save cash, it handles 4K timelines just fine.
• Double check that your power supply is at least 550W before installing.

To add to the point above: when you are dealing with 4K footage, especially the heavy codecs that come out of modern Sony cameras, you really want to look at the hardware decoding capabilities of your GPU. In my experience over the years, you basically have to go with Nvidia. You cant go wrong with them for Premiere because the software is just so heavily optimized for their architecture. For your $450 budget, just get a mid-range card from Nvidia's latest series. Don't overspend on the highest-end models because you hit diminishing returns fast. The main spec you want to look at is the VRAM. For 4K timelines, 8GB is the absolute baseline, but honestly, try to find a card with 12GB of VRAM within your price range. That extra memory pool is what keeps your system from choking when you start adding effects or color grading. Also, a quick practical tip... if you are shooting 10-bit footage, even a beefy card can lag sometimes. I highly recommend learning how to use proxies in your workflow. It takes like two minutes to set up and will make your editing buttery smooth, even on older hardware.