At such a low volume your R&D costs make the price of the chip not matter at all. You'll need to spend a couple of hundred hours at least to iron out the bugs on a device from NXP or TI. Count on doubling that by going for a more obscure manufacturer. Say at $80 per hour your labour costs to pay for 300 hours extra to get the obscure chip up &running would be $24 per board. Don't think a Linux kernel / BSP from a manufacturer website is ready to be used in a product. They are not! Tweaking and bug fixing will be required.
However for 1000 pieces a SOM (system on module) won't make sense at all. It will be cheaper to do a full-custom design.
The problem with that kind of accounting is that you aren't looking at long term costs, since 250 units of sales now might become 1000 units 2 years later. You also need to look at the big picture of what you end up with after the project: if you put the effort into getting a proper low-cost solution working, you end up with a valuable design with a value-add of the cost difference per unit. You need to think of it as an investment: if 250 units only breaks even (between the added labor cost and the saved BOM cost), and 250 units is the per-year sales figure, that represents a break-even time of just 1 year! After that you have ($100 minus $35) per unit of added pure profit.
There is also nothing obscure about Allwinner, they are one of the biggest SoC manufacturer in the low end tablet market and almost every SBC worth your time uses either Allwinner or Rockchip.
In addition, results from working out a low-cost SoC solution doesn't just affect one project; the design (and surrounding software work) can be reused in future projects. Reducing the price can also mean more sales. You can already buy fairly usable tablets for <$100, so why would I ever buy a product whose main feature is a low end SoC and costs $200? What about in a year when your competitor comes out with the same product but with a price lower than your BOM cost?