Welcome to my Swiss Alps!
Since you are traveling by motorhome, can I safely assume that you will not be using the trains , buses, lake boats, etc. elsewhere in Switzerland to any significant degree?
That can make the answer easier. The forum moderators are the experts, and will probably send you to this page.
I always defer to their more knowledgeable answers.
That said, your idea seems like the right one one to me.
I don't know if there is any regional pass that would be a better deal.
<"As these trips are obviously weather dependent we'd ideally like to purchase when we arrive. Does this seem like the best way of going about this?">>
One comment on your proposed journeys.
If your knees are not in good condition, you will find the walk from First down to Grindelwald ...an 1130 meter descent .... to be taxing.
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Quite few years ago, when I was a bit younger than you, I walked down from Kleine Scheidegg to Grindelwald. Slight greater altitude change. I was not in good shape. My knees and legs were very very sore for 2 or 3 days afterward. I don't know your fitness level, so I'll just pass on my experience for you to consider.
And, in mid-May, it is possible that the upper parts of the trail might still be blocked by snow. That area gets a lot of sun, so it may be OK.
In passing, I might mention that the major Alpine passes generally do not open until early June.