In your Schematic I see a 10uF capacitor on the output but it is not listed in the BOM/Components. 

You also have 4 resistors shown in the Schematic but only three in the BOM/Components.  Is R3 needed?

yes

I see that the MCP1640 comes in 4 flavors - MCP1640/B/C/D but neither the BOM(Components) nor the Schematic is clear as to which one it is. 

The image with it showing has CJKK for the marking (great pictures!), but the MCP1640 datasheet only shows that package being marked with a Bxxx (Page 21). 

Are you using the MCP1640T-I/CHY, MCP1640BT-I/CHY, MCP1640CT-I/CHY, or MCP1640DT-I/CHY?

No idea, found in my old inventory last 5pcs. currently worldwide sold out. I go to redraw to MAX1724

Sorry , but i'm a bit confuse about the theory of this schematics , i may be wrong but how can you charge a 10F in less than 24 hours with such current ?. According to the datasheet BPW34 source only 50µA ? . Can you explain more your theory . Because i found a total charge in 4days. Thanks

short current from BPW34 is about 4mA, it can charge 20F in sunny day without problems.

I would love if this fit in a standard light bulb form factor, disconnected from mains. A wonderful solution to exterior lighting light pollution at night! 

that fine balance between the MCP's minimum turn on and the diode's conduction voltage when charging the cap is elegant.

What is the approximate cost for all of the parts?

BOM cost looks to be about $15-20 (plus shipping) for single quantity.

Brilliant idea. Could a switch be safely inserted in to the circuit to allow charging but not turn the light on when it got dark? Allowing you to store the 4 hours of power for when it was needed? I understand the realiabilty reason of not including one but I was thinking reed switch? 

I try add switch in next release

Help an amateur out here! I am thinking this is something even I could build. I would love to hand them out to people for kicks (my thinking is to do what upstageleft suggested and cast it in resin--clear bubble on top and dark cylinder below. But how do I go about ordering the PCBs for this? I am in the US and have searched for a few sites, but am not sure how to configure what I want. 1 layer or 2? And THEN it gets harder. :)

testing smaller boards,  put in resin is possibly but I have no experiences with resins

is possible add mirror/metal top outside tile? or meybe normal reflector.

or add https://hackaday.io/project/158857-body-heat-powered-flashlight

or add "Forever Torch" Flashlight Just Shake old idea.

 for enclosure i rather prefer https://www.instructables.com/LED-Zippo-Flashlight/

Great idea  @upstageleft. Totally agree. Would be keen to make a few of these for where I live as we have power go whenever big storms happen.

Rather than a sample jar, have you considered casting in clear resin? Also, you could put a neodymium magnet at the base and make it a throwie (or just easily mountable). Seriously, I would buy a bunch of those...

it is possibly but I have no experiences with resins

A beauty PCB :)

Why D5?

through D5 is capacitor charged and also disabling power supply booster

Very illuminating!