Where to start building this wifi video streaming circuit?
tweaking off-the-shelf hardware to reduce mass

In practice, people often find that they can get most of the advantages of full-custom systems with much less time and money by tweaking off-the shelf PCBs.

Even if I were convinced I needed to make a full-custom system,
my first prototype would use off-the-shelf hardware just to make sure the system as a whole "works" and I haven't forgotten some critical subsystem, even if the prototype weighs 10 times as much as I expect the full-custom version to weigh.

Most of the mass of a typical computer is in the enclosure and the connectors.

My second prototype would use mostly off-the-shelf hardware, tearing off the enclosures I don't need; desoldering the connectors and other heavy components for stuff I don't use; and desoldering heavy connectors and replacing them with lighter connectors or direct wire connection.

I hear "The Ben Heck Show" frequently demonstrates a similar process.
The "WiFi Booster Mod by DBS" has some pretty pictures illustrating desoldering heavy connectors and replacing them with direct point-to-point soldered wires.

Full-custom Linux PCBs

In principle you could minimize the size and mass of the electronics by designing your own full-custom PCB(s).

If I were convinced I needed to make such a full-custom system, I would start from some open-hardware motherboards that run Linux and tweak the design slightly for my third prototype.

old-school rant

Wow, someone complaining that the Raspberry Pi is "bulky and heavy" ?
Do you know nothing about the Cray-1 or the Cray X-MP that several people --
"Help Chris boot his Cray-1 supercomputer",
"Raspberry Pi versus Cray X-MP supercomputer",
"A Cray for $35",
etc. --
claim can be replaced by a Raspberry Pi?