I think you've got a couple of options at this point... not great ones, but some.
#1 being, you could look to pick up a used AMD7970 gpu, which would be below 100 (not sure exactly because of your location ---- in my area i can get one for 70£)
#2 if you will be investing yourself into PC gaming, will you actually give up your current system and take a full upgrade soon? If so, then you should aim to get a higher end GPU so that you don't have to "rebuy" a GPU when you decide to build another system.
That being said, Bottleneck issue, is, somewhat a gray area, no matter which GPU you have. You will always have issues in games which are heavy on your CPU, such as GTA5. But games that are lighter on your CPU, you'll benefit from having a better GPU, even under a bottleneck situation. This means, you could even aim for an RX570 or GTX1060 if it's in your budget, but of course, bottleneck will exist.
viable upgrades as to drop in, go as follows RX560 (minimal bottleneck), GTX1050ti (minimal to mid-moderate bottleneck), RX570/GTX1060 (mid-moderate to heavy bottleneck)
#3 just upgrading. Here is an example of what you could get that will last you about 2 -3 years of decent gaming. The weak point is the CPU. And, if you don't have a windows 10 license, you would need to get that too. There is also another option, which is going with 6th gen CPUs that will be compatible with windows 7 which makes it much more flexible, but then I would suggest trying to get at least an i5-6400 (slower quad core, but not extreme price hike). The 7th gen as one that i listed, will work only on win10. (this is a budget focused suggestion, of course it's not an AMAZING system)
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel - Core i3-7100 3.9GHz Dual-Core Processor (£99.49 @ Aria PC)
Motherboard: MSI - B250M BAZOOKA Micro ATX LGA1151 Motherboard (£68.85 @ Box Limited)
Memory: Crucial - 8GB (1 x 8GB) DDR4-2133 Memory (£60.85 @ Amazon UK)
Storage: ADATA - Ultimate SU800 256GB 2.5" Solid State Drive (£80.09 @ Amazon UK)
Video Card: Gigabyte - GeForce GTX 1060 3GB 3GB Windforce OC Video Card (£189.97 @ Amazon UK)
Case: Thermaltake - Versa H21 ATX Mid Tower Case (£33.99 @ Amazon UK)
Power Supply: Corsair - Builder 500W 80+ Bronze Certified ATX Power Supply (£47.87 @ Ebuyer)
Total: £581.11
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-09-11 13:16 BST+0100
in all, you're stuck to get a new system, but... price to go one this is pretty steep. You could even wait till later this year / early next year, because new graphics cards from nVidia are coming