What is a safe way to carry around (wrt ESD safety) a single-board computer in an enclosure like this?
I'm not an ESD aficionado so take the following under advisement only. 
 Seriously though, I wouldn't call your two sheets of acrylic an enclosure for the simple reason that it doesn't actually enclose. It's one sheet on top and another beneath.  

I would consider this to be one of the worst arrangements for ESD possible.  You have two charge carriers sandwiching the very item you wish to protect. Have a quick look at this childish but useful cartoon illustrating static build up with acrylic rods.  Static charge in itself is not a problem.  Birds take 400kV sitting on distribution cables without even noticing.  The problems begin when they try to step off onto a grounded metal ladder. It's potential induced current that can damage circuitry and fowl, as in touching the pcb to one of the acrylic panels and perhaps the other.  If one panel is at 10kV because it rubbed on the silk hanky in your bag, and the other is at 5kV, that leaves the other 5kV to flow through your GPIO pin.  Zap. Grr.

It's not sexy or the Zeitgeist, but I do what the military do. Cover it in metal. Use a full metal box with fitted lid.  Small openings for the connector ports and try to get a low resistance path from the grounds to the box.  Aluminium or die cast zinc are the most machinable by hobbyists.  That way any accumulated static charge on the enclosure cannot flow though the computer itself. It will perform a Faraday shield effect around the board protecting it like an air plane is protected from lightning strikes.

A side benefit is the thermal transmittance of metal.  Your heat sink is not huge. You could expose it carefully through a tightly fitting hole, but you could just completely enclose it and rely on passive transmission through the entire case walls. There are Raspberry Pi enclosures that rely on this cooling method although the Pi is     a   bit less power hungry.

Crazily, you could also just wrap it up in tin foil for infrequent transport.  Just make sure that you touch something grounded before wrapping it like a radiator or the screws on a PC. If you're concerned about ESD whilst assembling it (perhaps justified due to the acrylic), you could use an earthed wrist band like:-