DfT did not specify that 802s could not be used. What you may be referring to is that DfT specified an increase in capacity by December 2017 and then by December 2018, as shown here
https://assets.publishing.service.g...ta/file/431065/tsr-capacity-specification.pdf
It has been quoted that Hitachi was unable to meet that date due to capacity issues at Newton Aycliffe, whereas the fleet was actually built in Japan
https://www.railmagazine.com/news/network/first-hitachi-built-train-for-transpennine-express-arrives-in-uk#:~:text=Built in Kasado, Japan, it,Pistoia in Italy, and Kasado.
The inclusion of the Mk5 and 397s in the bid was intended to fulfil these timeframes, albeit by the time the contract was agreed 6 months later the December 2017 date had been missed. A consistent order of Mk5s and 397s would have been compliant to run express services on the whole TPE network on the basis of the requirements specified in the ITT.
The reason FirstGroup ordered the (much more expensive) 802s was because, over and above the Invitation to Tender, their bid included revenue generating paths on the East Coast Main Line; an additional train to Newcastle, and one of them extending onto Edinburgh.
The way that ORCATS revenue is allocated then meant they would get 2/5ths (rather than 1/4) of the walk-up revenue between Newcastle and York, and 1/4 of the revenue between Edinburgh and Newcastle, irrelevant of how many people actually used their trains. This is why they offered those services over and above the specification, because it made commercial sense.
In order for ORR to grant them the track access and Network Rail to grant the contingent rights (they were never awarded firm track access rights for those paths), they needed 125mph capability, rather than the 100mph offered by Class 68s.
These commercial factors are what made FirstGroup push ahead with the December 2019 timetable change even though they didn't have the drivers trained to do so. The trains were booked to run and so TPE were allocated the revenue.
The reason some parts of TPE tried to delay acceptance of the Mk5 fleet is because it cost more money without this additional revenue being generated, it was just additional cost both due to leasing the carriages and having to train the drivers.
The Nova 3s in themselves have been quoted as having good reliability statistics when there was a reasonable number of them in service. Blaming poor build quality / the cracking issues is an interesting one when you consider that the Hitachi fleet has had a similar issue.
www.modernrailways.com
It is worth considering why FirstGroup made decisions that, on the face of it, make only commercial sense. This is because, even pre-May 2018, they were losing money on the TPE franchise and expected to lose over £100m over the life of the contract.