I got my thermal camera, so the results are in! It looks like the peak temperature I have managed to get for the transformer is 129°F. Here are some pictures. The first shows the temperature of the transformer. This is when the power supply is producing 20mA at 200V with 12V in:

Next up, the controller:

And next the MOSFET:

Is that OK? Well the data sheet for the transformer says:

It is recommended that the temperature of the part does not exceed +125°C under worst case conditions. 

125°C is 257°F, so we are well under that. These measurements were taken with an ambient temperature of around 70°F. We could expect worse behavior in a closed case, but I think we may still have some headroom, so it could be that the current sense resistor value that I chose could be reduced some. I will try this with a 0.04Ω resistor and check the temperatures again.

I will list the final component values in a moment, but for now here are some performance results:

12V
Rload (K)	Vout	Iout (mA)	W		Vin	Iin (mA)	W		Eff
20	199.6	10.40	2.08		11.96	238	2.85		0.73
15	199.5	13.70	2.73		11.96	298	3.56		0.77
10	199.2	20.00	3.98		11.94	415	4.96		0.80
20	177.6	9.26	1.64		11.96	195	2.33		0.71
15	177.5	12.18	2.16		11.96	243	2.91		0.74
10	177.4	17.84	3.16		11.96	347	4.15		0.76
5V
Rload (K)	Vout	Iout (mA)	W		Vin	Iin (mA)	W		Eff
20	192.4	10.00	1.92		4.85	500	2.43		0.79
15	172.6	11.84	2.04		4.84	502	2.43		0.84
10	141.0	14.17	2.00		4.8	508	2.44		0.82
20	177.0	9.22	1.63		4.85	500	2.43		0.67
15	172.5	11.81	2.04		4.84	502	2.43		0.84
10	141.0	14.18	2.00		4.8	508	2.44		0.82
3.7V
Rload (K)	Vout	Iout (mA)	W		Vin	Iin (mA)	W		Eff
20	174.0	8.80	1.53		3.6	373	1.34		1.14
15	154.0	10.34	1.59		3.6	370	1.33		1.20
10	130.0	12.80	1.66		3.6	364	1.31		1.27
20	172.0	8.80	1.51		3.6	380	1.37		1.11
15	153.0	10.30	1.58		3.6	373	1.34		1.17
10	129.0	12.70	1.64		3.6	370	1.33		1.23

For each input voltage, there are two sets of numbers. The first set is for a V_out limit of 200V. The second set is for a V_out limit of 177V. Here are some takeaways:

1. My measurements for 3.7V are clearly wrong somewhere, because they give an efficiency of > 1!
2. Ignoring that, we are hitting efficiencies of between 70% and 80%, which is pretty good!
3. We can get 20mA out of it if we put 12V in. We could probably get more, but my test equipment (such as it is) can't go lower than a 10K load without starting a fire. In fact, the thermal camera shows that my 10K load resistor was hitting 400°F as it is! Anyone want to buy me some lab equipment? 20mA will power just about any 6 tube clock.
4. On battery, we can get 8mA out, which is enough for 4 to 6 small tubes. Which is fine.
5. On USB (aka 5V), we can get roughly 12mA out at a decent voltage. That is good for a lot of tube types. If I can decrease R_sense, so I can get more current flowing, we could improve these numbers.
6. The voltage regulation starts to fail as load goes up and input voltage drops. Again, I know from experience that if I can drop R_sense, I can get a more stable output voltage

So here is the final circuit diagram:

1. The MOSFET is an AO4294. This has a 100V V_ds and a very low R_ds(on). Other options I tried tended to get a bit warm.
2. R_sen is 0.06Ω.
3. R_freq is 47K.
4. R_snub is 249Ω
5. C_snub is 360pF.
6. The diode is an ES1FL R3G.
7. The transformer is a Wurth ER11/5 - order code 749196141.

Here is a picture of the board layout: