Convert a digital design based on arduino into VHDL code
Why do you want to use an FPGA?  The requirements you've mentioned above don't give any hint of anything with sufficient processing and parallelism demand to suit an FPGA.  In fact, Bluetooth is a massive endeavour in it's own right, even in software, I wouldn't fancy trying to build a Bluetooth stack in VHDL, even as an educational exercise!

If you want a cheaper Arduino, redesign the PCB to use a cheaper, but compatible, device, leave off the bits you don't need, integrate any new bits you'd like.  Even that is a lot of work (especially if you want to create a fully-validated product which you can CE-mark, get FCC approval for, etc.).

And the next step might be to use an even cheaper micro, with all the extra software effort that that entails.

But I wouldn't consider an FPGA for this job.