The script defaulted to them bickering as part of Screenwriting 101: instant conflict.
"The Five Doctors" was produced for the show's twentieth anniversary in 1983 and was an interesting mess Tom Baker refused to return to the show, so they used footage from the then-incomplete and shelved story "Shada" so they could have the Fourth Doctor[...]
The Five Doctors Archives
Now as a member of Rifftrax, he's doing a live show in August, a comedy show built around the 1983 special episode of Doctor Who: "The Five Doctors." We chat about riffing, respect for Doctor Who, and being known as Tom Servo.
"RiffTrax Live: Doctor Who – The Five Doctors" will be shown on Thursday, August 17th at 8:00 p.m[...]
He chose this theme to make the special stand out rather than just end up with a big multi-actor crossover that's generally a game of musical chairs when writing the script, which was what happened with The Five Doctors back in the 1980s, which didn't have any real emotional impact despite the fun and games[...]