Although you are correct in that the Xeon E3-1235L v5 and Xeon E3-1240L v5 have the lowest rated TDP (both at 25W), I would recommend staying away from these processors.
One thing to understand straight away is that under most conditions, the actual power consumption of the process will not be equal to the rated TDP. The rated TDP only measures the maximum thermal output the processor is capable–that is, under full load. For example, the i7-6700K is rated for 91W, but only consumes 3.4W at idle. Note that this does not necessarily scale with rated TDP: the i3-3240 and i7-3770K have vastly different rated TDPs, but both consume roughly the same amount of power at idle.
One thing you've noticed is that L and T series processors can consume significantly less power, even at idle/low load (although I'm not entirely sure what Intel does under the hood to do this.) However, this difference isn't necessarily huge–especially given their lower performance due to lower clocks.
Given the nature of what you're doing, I wouldn't expect the CPU to be under full load most of the time, and therefore, I don't believe it makes sense to buy an L or T series processor given...
The cheapest L series v5 Xeon (the E3-1235L v5) has an MSRP of $250.00 (if you can find one at all.) Meanwhile, the Celeron G3900 is only $30.61 with a rated TDP of 51W. At the US average of $0.12/kWh, even at full load, it would take roughly 8 years of continuous usage to offset the cost differential between the two processors. It simply doesn't make sense to buy a more expensive processor.
The final nail in the coffin is that v5 Xeons simply aren't compatible with motherboards with consumer chipsets–including B250. While you could get a lower TDP consumer CPU, the price differential (a delta of $67.38 as of right now for the cheapest T series processor, the i3-7100T) probably isn't worth it.