Wow, sounds like a fun project!
I think that's something that would be doable on-chip with an Espruino device, but you may need to carefully craft some C/assembler code to handle the ASIC4 protocol.
As @allObjects says, something like the MDBT42Q boards/modules might be an option. It's not WiFi, but is very low power and would allow you to create a Bluetooth LE connection to a phone/desktop where you could use the phone's WiFi connection and storage?
The nRF52 used for Espruino's Bluetooth boards has some neat tricks up its sleeve too. The on-chip peripherals can be 'glued' together a little like with an FPGA - see http://www.espruino.com/NRF52LL
I'm not sure if it's good enough to use to decode/recreate the 12 bit packets (I doubt it) but it might be handy. It'd be ok if all you needed to do was receive, but handling the transmit side requires a very quick response.
Is the data always clocked out at 3.84MHz? It strikes me that the easiest way to handle it might be to just use Espruino's inline assembler. You could just arrange that your function gets called when the CLK line first changes state, then you could read in or output the bits of data one at a time, just inserting the odd
NOP (no-operation) command to pad everything out until the 12 bit packet had finished.
Realistically in order for you to not go completely insane you're going to need to get a digital storage scope though so you can see what's going on!
... having said all that, small FPGAs are quite cheap (or CPLDs - basically the same but they definitely remember what they were programmed to do when powered off - not all FPGAs do). Decoding 12 bits of data is actually something that would be a really fun and instructive project to do with one of them.