Takayama is a way out from your other destinations. I personally thought Takayam was overrated - the old streets were mostly souvenir businesses, and there were just too many tourists around to enjoy the atmosphere. Similarly, Beppu is bit further away; I haven't been myself so can't comment on it specifically.
Most accommodations open their bookings 3-6 months ahead, so you're definitely too early! If there is a particularly special accommodation, e.g. a small ryokan, you could try e-mailing them ahead of time to see if they'll take an early booking (I've has success with this in the past).
Beppu being that far to travel is why I would seriously consider either dropping it from the trip, or seeing other things close by. That's a whole day spent just in transport! You could spend the time given to Beppu in other places - e.g. Kanazawa (between Takayama and Kyoto) or Miyajima (always worth a day trip from Hiroshima).
I would highly recommend using the luggage forwarding service. I bring both a large and small suitcase, and when I have a few days between larger cities (e.g. your time in Takayama), I forward my large suitcase to meet me a few days later, and take a small bag with me on the few days. Much more convenient for travel, especially with the new rules around taking large baggage o the shinkansen. Your hotels can assist with the forms and payment for luggage forwarding; I've always found it very easy. It's also quite reasonably priced - usually about 2000 yen for me to send a big suitcase from Tokyo to Kyoto. It also means you have a spare bag to pack all the souvenirs in at the end of your trip!
I would recommend reserving a pocket wifi before you leave, to ensure you will definitely get one (i.e. they don't sell out!). I usually go through the following website: https://www.gmobile.biz/sbinbound/
You can organise to pick it up from the airport post office; if you arrive after the post office is closed, it can be sent directly to your hotel.
Note that some accommodations may be outside your current budget - ryokan can easily go over $100 per person, as can decent accommodation in cities like Tokyo and Kyoto. Perhaps be a bit more flexible in that part of your budget. Also make sure you have run the calculations to make sure a 21-day nationwide JR Pass pays off - sometimes it's cheaper to not cover your whole trip with one pass, and buy separate tickets, or use a combination of smaller regional passes. This takes time to work out, but you have plenty of time!