The biggest interest you have in this is not going to court over it - that way madness and a serious loss of money lies. So it is good that you have been able to discuss with your neighbour. I would recommend doing almost anything else other than getting into the sort of argument over it that ends up in court. On the other hand, your need for a fence for your dogs is a very reasonable reason for you to try to sort out the issue.
I would start by talking to, or dropping a note around to, your neighbour just to let him know what you are proposing - eg "As you know, I have dogs, and I need to put a fence around my property in order to keep them safe and stop them from escaping. I'm going to have a formal survey done and I'll let you know if there are any issues. Thanks". (NB Even if you've had the survey done I'd suggest not presenting your neighbour with a fait accompli at this stage.)
The follow up will be "I mentioned to you that I was going to have a survey done of the boundary between our properties before I put in a fence for my dogs. The survey shows that a very short part of the lowest retaining wall that I think you had put in is in fact on my property: I enclose a copy of the survey showing how the retaining wall relates to the proposed fencing line. The new fencing will obviously need to take account of this small part of the retaining wall in some way: I hope we can discuss and agree how this can be done. I will of course organise and pay for any necessary work."
After that it's a matter of negotiation between the two of you. The fact that you are happy to connect the drainage at the bottom of the retaining wall is a plus as it will remove one of your neighbour's possible concerns. That leaves the question of whether the retaining wall should stay where it is with the fencing cutting across it, or whether it should be relocated along the surveyed boundary, and if it is to be moved who is to do the work and who is to pay for it.
If your neighbour has been in his house since the 1970s then he is 1) elderly and 2) probably disinclined or unable to put in much effort to making major changes to "his" property. As it is a relatively short length of retaining wall and not very high I think that the costs of you taking on any necessary work are probably minimal compared to the potential trouble of trying to get your neighbour to do it.