Target Display Mode using 2019 MBP and 2012 iMAC I currently have:


2019 MBP on MacOS 10.14.6 
2012 iMAC on MacOS 10.14.6 (thunderbolt 2)


I am connecting them to each other using a Thunderbolt 2 to Thunderbolt 3 adapter, both of which are brand new.

I was able to trigger target display mode ONCE. Then cmd + F2 stopped responding entirely.
Things I've tried: 


reset the PRAM/NVRAM on both devices 
cmd + F2 + fn
every combination of the ports on each end 
booted the iMAC in safe mode (can't do it on the MBP; it's a work laptop)
tried initiating TDM by connecting iMAC to MBP 2014, still no dice on cmd + F2


I've perused all the Apple support pages and similar stackexchange questions to no avail and now I'm at my wit's end. Did anybody face something similar and was able to resolve?
Did you use a 'Thunderbolt' or 'Thunderbolt 2' cable to connect the two macs?
Also judging by your description the iMac has Mojave installed whereas TDM would work only on an iMac with High Sierra or below.
For reference please follow this official Apple article: https://support.apple.com/en-us/HT204592