I wasn't around last night to see this thread and this morning it makes for a very disappointing read ... not so much for the fact that the dove perished, but because the responses the OP received when asking for advice, were so devoid of empathy for the fate of the dove.
That saddens me, because it appears that for at least some members of WAB, as far as 'wildlife' is concerned, the 'wild' continues to be of more interest to them than the 'life'. A great pity then that k4t3 only got the benefit of input from those individuals.
To me at least, it is pretty clear that there are only two courses of action in a situation such as this:
a) do nothing ... and content yourself to be a bystander to the often harsh workings of nature. It is clearly not the right course of action to interfere with any bird of prey (or another natural predator) after it has brought down a dove. Perhaps too in this case, it was unwise to remove the bird from imminent predation by the cat, which would surely have quickly ended its suffering, and had it been a feral animal might well have also been in dire need of a meal ... however that is mere conjecture.
b) remove the bird from immediate harm ... this is what k4t3 chose to do, before asking us for advice. However, whenever you choose to interact with nature in that manner, you immediately put yourself in a situation of dilemma, both morally and, let's not forget, legally too. By taking an injured bird into your care you temporarily assume legal responsibility (under the Animal Welfare Act 2006) for its welfare as a 'protected animal' to an extent equal to that placed upon you by ownership of any animal.
At that very point therefore, the dove became a 'casualty' and rightly deserved to be considered and treated as such. All other conjecture therefore should have become secondary to the immediate welfare of the dove.
Now it is obvious that not everyone will possess either the wit or the ken to know how to assess the extent of the injuries, or to deal directly with them ... however, whilst recognizing our own limitations, it is just not enough to believe 'there is nothing more I can do' is the same as 'there's nothing more that can be done' for any casualty animal.
Every registered vet in the UK is required by law to operate an out-of-hours contact number, and to provide appropriate treatment in alleviating the suffering of any animal (including wildlife) presented to them. So a quick call to the local vet might have been the best suggested course of action.
The vet might well have euthanased the bird, but only after a proper examination of its injuries ... and for the bird the end would have then been both quicker and comparatively pain free. More likely than not, this procedure would also have been done without charge.
If the vet had a close working association with a wildlife rehabilitator, then perhaps he/she might have concluded that the bird could have been patched up, fluids replaced, its wounds dressed, antibiotics administered, and then turned over to the rehabilitator to oversee its recovery and eventual fully recuperated return to the wild.
Please then, when giving advice to someone on this forum seeking advice on how to deal with a wildlife casualty, try to ensure that ALL the available options are considered.