Some tantalum capacitors used to die with actual FIRE leaving crap on the circuit boards. Modern tantalums probably don't behave like that anymore. Like they said, they have problems with overvoltage.
I wouldn't use tantalum capacitors unless I need something that tantalum has, like a predictable low ESR across some temperature ranges or whatever (higher than ceramics and not prone to cracks or microphony or voltage bias, and lower than esr of small capacitance electrolytics, which is usually around 0.3-0.5 ohm and up for < 100uF).
Electrolytic capacitors are often cheaper and you can simply use capacitors rated for higher voltage (like 63v-100v) because the diameters and heights of such small value electrolytic capacitors are about the same (usually around 5mm diameter , 8-11mm tall) ... and if height is an issue, or you don't want to use surface mount electrolytics, you can place regular through hole electrolytic and lay them flat on the board and optionally - if you really are paranoid - use a drop of glue or for example that thread locker to hold them in place.
You already place the 32 kHz oscillator for the timer chip flat on the board, so it looks to me like you have no issue with that, and there's plenty of room on the pcb. There also doesn't seem to be any requirement for surface mount only (for example if you want the back of the board to be perfectly flat) as there's lots of through hole parts like the headers and the holes for additional regulator.
Anyway... don't see the attraction of that INA219 chip ... doesn't look that special to me and it's expensive, at around 1.6$ if you buy 100. It has only a 12bit ADC and not a very good one at that.
You could get a decent 16bit ADC with internal voltage reference like let's say LTC2472 for 2$ (if you buy 100) and you get a 16 bit value every ms or every 4 ms depending on how you configure it... and you put a 0.5$ microcontroller to read the data from SPI and calculate power and all that other crap and push out the values through i2c or whatever you use.