$300 to spend on a tripod and ball head. What would you buy??

I bought the next step up in the tripod as it had the larger diameter legs and is rated at 80lbs+ so a bit of overkill involved. Just a little bit over your budget. Different brand same tripod

https://www.amazon.com/Carbon-Tripod-Professional-Birdwatching-Adapter/dp/B07H8V8WWM/ref=sr_1_3?dchild=1&keywords=innorel&qid=1625103377&sr=8-3

I also have an inexpensive ball head with the larger 45mm ball that werqs just fine

https://www.amazon.com/Professional-Tripod-Release-Camera-Monopod/dp/B0719GTW4N/ref=sr_1_4?dchild=1&keywords=ball+head+45mm&qid=1625103993&sr=8-4

Although I picked up a nice gimbal a week or two later. The gimbal is so much smoother than a ball but it is bulkier and heavier as well so a tradeoff.

https://www.amazon.com/Neewer-Professional-Panoramic-Arca-Swiss-13-6kilogram/dp/B01M262LLV/ref=sr_1_4?dchild=1&keywords=carbon+fiber+gimbal&qid=1625103634&sr=8-4

There are plenty of picatinny to arca and back adapters around also.

https://www.amazon.com/s?k=picatinny+to+arca+adapter&ref=nb_sb_noss_1

So many choices out there there are.
 
Nightsnipe is a great choice IMO for a ball head/tripod. For under 400, the hunter elite combo has been a solid performer for me. The head itself holds the big bore steady as a rock via pic adapter. Added a BOG grip for other rifles. Smooth locking knobs and not bad weight wise to lug around. Removable leg for bipod use too. 


https://www.predatorhunteroutdoors.com/index.php/product/hunting-tripod-ball-head-nightsnipe-ns67/

https://www.predatorhunteroutdoors.com/index.php/product/hunting-shooting-tripod-carbon-fiber-nightsnipe-tournament-hunter/
 
I would save some more money.

I would be thrifty and purchase cheaper tripod legs if you don't have to carry it to far and go with the Slik legs linked below for $100. They are really solid legs for $100! 

https://thermalopticsplus.com/collections/mounts-and-tripods/products/slik-pro-700-dx-heavy-duty-tripod

I would splurge on the head. You will never regret it. The Really Right Stuff Bh-55 Soar is an awesome shooting and photography head. I love mine! I have it combined with a carbon Gitzo tripod. The Anvil 30 is another popular choice but less versatile in my opinion. 

https://thermalopticsplus.com/collections/really-right-stuff-tripods-ballheads-and-accessories/products/bh-55-lr-soar-series-ballhead

This combo costs about twice what you are looking to spend. However, it is absolutely worth saving your $$$ for and doing it right the first time. If you don't like the head you will be a able to sell it easily for little loss of money. RRS heads are very desirable on the used market.