Have you considered renting or bringing a wheel chair for your dad? It may help him when he gets tired. It may not be easy everywhere, but when using it where possible, it would enable him to save his energy for the places which are a bigger challenge. This is what my friend has been doing for years; she can't walk a lot due to her illness.
We took a wheel chair for my son after he broke his leg. Of course that is not the same situation, but we flew to Rome the day after his cast had been removed - he had been in that cast for almost 12 weeks so he couldn't walk at all and he wasn't allowed to walk for a few more weeks. Just a thought.
I may have missed what your dad's (family's) preferences would be. Of course you know that the options you mentioned will make for different experiences.
If you could travel early in October the cruise may be an option. Since you are taking an Eastern Med cruise in June, I am assuming you would do the Western Med with your dad. Personally I wouldn't do that later in October, but many would disagree. (Actually I would opt for September myself if possible.)
If you would want to travel to other regions, you might want to fly or take a train from Venice. Flying to Sicily is easy and quick.
If you are up to driving, you may consider to drive from/to Florence or the Amalfi Coast. I know many here say never to drive in Italy, but if you rent a car from the airport, you skip driving in the cities. You can then drive to the airport of your next destination and take a taxi or limo from there. Florence to Rome is a 3 hours drive and Rome Airport to the Amalfi Coast is also a 3 hours drive (last time we were in Italy - the trip with the wheel chair - we flew to Rome, stayed at the Hilton Airport hotel, picked up the car the next morning and drove to the Hilton Sorrento. Took 3 hours in total.). If you start driving around 11am, you will notice that the highways will start to get empty. Italians will start their siesta, and driving will be very easy because you will feel like you are almost the only one on the road. The 3 hours needed for the drive will be exactly during siesta time.
In my opinion, that would be the most comfortable way of travel for your dad, provided you have a car with enough space. While at the Amalfi Coast, you could take private transfers. (Of course you can take private transfers to transport you to and from various city locations, but with 4 adults, you have to take into consideration the amount of luggage and such. You will probably need to have a minivan with driver to be comfortable.)
Staying in Rome of course would be the most comfortable option and unless this was to be a once in a lifetime trip to Italy, this would be my personal choice. As mentioned earlier, if you have a week, you have to take into account the days or arrival and departure, jetlag, a day at the Vatican, and I agree that it is so nice to have some time to actually enjoy the surroundings and to actually feel the atmosphere. But then again, last time I was in Italy, I was told by an American that this is "the European" way of vacationing.
I don't think that there aren't any bad options as they are all great. It's just that we are all so different.