If you are still inclined to the 70-300 af-p, the non-VR version sells used for $99-127 good-like new condition on Amazon. Or look at the af-s(compatible with D5200) 55-300 used in very good condition for $167. Links below, make sure to swap to used for those prices.

Nikon AF-P DX NIKKOR 70-300mm f/4.5-6.3G ED Lens for Nikon DSLR Cameras https://www.amazon.com/dp/B01KJ8URUK/ref=cm_sw_r_apan_i_Y0NHWR7348JQ4G72B434

Nikon AF-S DX NIKKOR 55-300mm f/4.5-5.6G ED Vibration Reduction Zoom Lens with Auto Focus for Nikon DSLR Cameras https://www.amazon.com/dp/B003ZSHNCC/ref=cm_sw_r_apan_i_JP93DNVSHGHN336318ME