I almost always recommend the shinkansen over domestic flights but in this case might go the other way. This assumes that you will not be doing any other long-distance train travel during your stay so will not have a JR Rail Pass. If you WILL have a Rail Pass, I'd still go with the train.

Assuming no Rail Pass, the shinkansen to Kyoto would add significant cost over using Avios - or any other points/miles - to purchase the flight. Since you will already be at the airport, I think the convenience of flying from HND to ITM outweighs the benefits of the train. As mentioned above, ITM is not in or near Kyoto so you will still be taking a limo-bus to Kyoto Station.

Also as mentioned above, in Japan luggage can be a concern. If you intend to take much luggage, the flight option also has a benefit. You would have to handle it from Arrivals to Domestic Departures at HND but it will be checked on to ITM. The limo-bus from ITM to Kyoto will be far better equipped to handle your bags than the shinkansen, which has limited bag space.

For the trip from Kyoto to Tokyo, I agree with the above suggestion to ask your hotel to have bags shipped ahead a day in advance to your next hotel via the excellent "takkyubin" forwarding service, keeping what you need for your last day/night in Kyoto in a small overnight bag. That way you don't have to wrangle baggage through the train station and onto the train, which has no checked bag service.

For traveling in Japan, the minimum amount of luggage one can get by with is always a good idea.