Hi
I am very sorry that you are finding yourself in that situation. It is usually the most difficult when you as a loving owner come close to the time and want to get it right but are very torn between various competing factors and emotions in play.
Personally, I am always happier and relieved if my especially my older piggies can make the journey to the Bridge from their usual surroundings at home but it depends on whether they are in distress or pain at some stage except for the very last half an hour or so when oxygen deprivation can hit but a piggy is pretty much no longer conscious and with it, whether a slow dying process hits a bump but with no recovery possible by that stage or whether they are likely to pass away on the way to the vets or in the waiting room and it is too late to take them etc.
These are what I call 'circumstances', which are not necessarily in your control and which you have to judge as you go along. A slow natural death is an individual process, depending on the illness, the order in which the organs break down and how healthy they are; each passing is unique and its own dynamics. Monitoring a natural death can come with quite a lot stress, heartache and self-examination along the way; especially if you suffer from anxiety because it is a physical process, some of which can be distressing that you cannot hurry on and there is comparatively little you can do. It can take days and it is not necessarily a gentle fading away.
I still prefer this way, which about just over half of my piggies have been able to take over the years but be prepared to change your mind if needed. It is less stressful for the piggies but it can come with its own challenges for you.
PS: I am moving this thread to our End of Life and Bereavement Section for ongoing support. Please bookmark the thread so you can always pick it up easily but we can let it run for as long as you need and be there for your own needs as much as for practical questions for Kevin and his mate.
If you struggle, you may find this link here helpful so you know better where your own feelings and priority lie and can manoeuvre the whole process with a clearer mind (but not a less aching heart):
Operation, Terminal Care or Euthanasia? - Helpful Questions to Ask Ahead or in Hindsight
With Kevin's extremities being rather cold, it is likely that his body has already started to close down and that the blood circulation is no longer working optimally. Please keep him warm but not hot and always give him an option to move away from a source of heat if he becomes too warm.
A Practical and Sensitive Guide to Dying, Terminal Illness and Euthanasia in Guinea Pigs