honestly, you are on a highway to nowhere with this.
the washer was to be repaired under an extended warranty, this suggests that it is outside the manufacturers 1 to 3 year warranty period.
the handyman has fcuked up, there is no denying that, however what he has damaged is a faulty machine.
the value of a faulty machine is roughly £0 as it serves no useful purpose and you would have to pay to have it removed.
the original engineer told you that it needed to be removed in a certain way to prevent damage, you have in there minds gone against recommendations and damaged the machine. (even though it was not you that did it, it was someone under your instruction)
If indeed the space was too small for the machine, then it would void any warranty from either the manufacturer or the extended warranty company.
one possible route may be the people who originally installed it, however that is a long shot in the extreme.
possibility's,
1
Take the extended warranty company to court,
they will argue that they did not damage it, told you to have it removed from its position in a safe way and they would return to fix it.
chance of winning any pay-out 0%
2
take the handyman to court
he will argue that you asked him to remove machine so others could work on it.
may even accept that you said remove side of cupboard to do it.
his options are
lie and say he just asked me to move it, no mention of not pulling it by door
suggest that a properly installed machine should be removed in the way that he did it and any damage was caused by incorrect installation, making it impossible to remove without damage.
say it was impossible to remove cupboard side without first removing machine as the fixtures were screwed in next to the machine and he couldn't access them.
chance of winning any pay-out 10% at best
3
approach the handyman's insurer directly.
a) he is not insured -quite possible if you didn't ask or check first.
b) they will argue that the loss is insignificant because of the age of the machine and the fact that it was already broken.
c) blame it on the original installers because a machine should be installed in a way that allows for removal encase of required maintenance.
4
take original installers to court
a) may not have worked for you, could have been previous owner or tenant
makes it hard to get anything from them
b) was well outside there complaints time, if there was a problem then it should have been reported at job completion, not x years later.
c) if you have not got proof that they did the work, could easily claim that they were never there.
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My suggestion, If the same thing had happened to me,
Buy a new dishwasher and have it installed
rethink who you have come and do work for you and check there qualifications / insurance etc.
check, recheck and double check the space you have before purchasing something to fit in it, have it listed in the order when you buy a machine, (must fit in space W600 D824 H582)