I had this problem when my older dog was alive. She was 14 and suffered from arthritis like your golden and the other one was still 12 but with no arthritis and still quite fit and active for her age so I was torn between exercising Jess enough and not over-exercising Mist.

If you have somebody else to help you, maybe slightly overlapping the walks in some way, ie, bringing the young dog out of the back door as somebody else takes the older one into the front door after a little stroll for him or vice versa so that they don't see each other coming and going. He won't see her leave then and it might help.

I would just take Mist for a little stroll around the block first so she could do her business and just move her legs a bit and then I'd take our Jess for a longer walk afterwards. Mist had to be put to sleep at the end of January though so I only have Jess to worry about.

So, try and take your older dog just for a little stroll on the fronts to do his business and move his legs a bit then as you bring him in one door, have somebody else take the other out of the other door so they don't meet or see each other. The older dog will be blissfully unaware then that the younger dog has been taken out, lol.
If the other person can't actually exercise the dog for whatever reason, you can sort of swap places again outside once the older dog is inside and the other person come back inside whilst you resume on your longer walk with the younger dog.

I thought ideas I came up with might help you out.