That fixture is easily removed and replaced with pot lights by ant decent electrician It is done all the time IMO the depth of the raised ceiling will be an issue since the light has no where to go to light up the whole space so if you do not want to drop the ceiling then you will need lighting in that box andoutside the box so under cabinets . What exactly in in the perimeter over the cabonets ? IMO remove it all to the ceiling then you can light the space properly

We had the same situation and took the edge trim off and replaced the fluorescent with two round led lights.

I would do what mojavemaria did + add a light over the sink in the soffit and undercabinet lighting.

Those two led lights are an OK option for area lighting. They'll create a shadow from your body wherever you work at any counter, sink or stove. Some undercabinet led lighting can help. You'll still need 450+ lumens at the counter task lighting for work zones. I'd look into tearing out the drop back to the cabinets. 4" recessed cans then can be rlaced inward above the counter edge and above the sink and stove.

For your $300 budget, you can replace the fluorescent fixture with a semi-flush fixture that holds 3 LED bulbs. Use 75-watt equivalent LED bulbs to get enough light. A semi-flush fixture will extend most or all the way out of the recessed area, so light can go sideways as well as down. See examples:

https://www.lampsplus.com/products/close-to-ceiling-lights/usage_semi-flush-mount/number-of-lights_3-lights/