Color me a nerd. I got my transistors, a couple days ago. And since it is Friday night, I obviously have plenty of time to play with transistors, lol.
I have wired up the FET vs BJT challenge.
Red corner:
ZTX618
NPN transistor.
Blue corner
BSS816NWH6327XTSA1 (this just rolls off the tongue, don't it?)
This is the "PFET" which I suggested earlier, based on a 30 second Mouser search, sorted by price. And apparently, I can't tell the difference between a PFET and an NFET, cuz it's an NFET, lol
So I put 200R resistor on Collector/Drain. And for the NPN, I put a 2K resistor on the base. Gate, direct connection. Gate/base biased with Vsupply.
Red corner - 30 cent BJT
@ 2.4V, my PSU shows 9mA (experience tells me this reads low, but it verifies circuit working as intended) and my multimeter (Cheapo Uni-T pen meter) shows Vce 0.004V, Vsupply 2.396V
@ 1.8V, my PSU can't decide if it's 4mA or 6mA... and penmeter shows Vce of 0.004V
Blue corner - 8 cent FET
@2.4V, 8mA, Vds 0.001V, Vsupply 2.396V
@1.8V, 4ma, Vds 0.001V
The BJT is impressive to me, but the numbers don't lie. FET still wins, here. Well, THIS particular sample, anyway. Maybe next sample of same FET not as low. I'll have to try the BJT upside down, next.
Real beauty of FET for low frequency power switching, though... can run them in parallel ad infinitum to decrease voltage drop even further while increasing max heat dissipation. BJT seems to bottom out, as per datasheet, 4mV is the drop at even 100mA.
So, anyhow, OP has many options. Yes, you can switch without a diode drop using BJT or FET.
* Using 29.3 ohm load resistor, to give ZTX room to shine, Vsupply 2.31V (voltage drop from long leads!), and 200R base resistor, Vce is 0.009V. So it is not meeting spec at this voltage. (This is only 78.8ish mA, but I didn't have right resistor to get 100mA.)
Same load resistor and supply voltage, NFET has Vds of 0.009V. TIE!
At 2.3V, BTS FET is right on par with graph in datasheet at this point. ~9.46mV drop is what would be expected per the graph. I'm not inclined to think this is particularly "good" example of this FET.... even though it outperformed at 10mA, the graph doesn't scale down very far. But maybe the BJT is a bad one...