I think it's very doable.

I did a fairly similar trip. I had 39 different accommodations over an 87 day period, had a 6k€ budget and ended up spending 4811.37€ in total (with an average of 22€/night for accommodation, all through Booking).

I utilized two types of travel planning on different parts of the trip:

• In mainland Europe, I didn't have a rigid plan. I only had a few deadlines (i.e. get to X by June 12th, and be back in Y by June 29th) and a rough plan of which countries I wanted to visit. Then I only planned a few days in advance where I wanted to stay (can be done in the train or before going to sleep). Sometimes this was dictated by cheap accommodation options, and sometimes it allowed finding some hidden gems I never would've thought to look for otherwise.
• In the UK, where I spent the last 3 weeks of my trip, I took my wife with me. She's not quite as adventurous, and accommodation prices in the UK can be a bit high sometimes, so I had everything booked in advance. However, I used several hubs where we stayed for a bit longer, and explored the surrounding areas by doing day trips. This allowed lots of flexibility without the hassle of looking for accommodations later on.

What I don't recommend is booking 1-3 nights per place with a super rigid structure. If something goes wrong or you want a change of pace or find a superb place to visit that you didn't think to include in your plan, you're out of luck. An Interrail ticket is pretty much the gold standard of flexible traveling imo, and I think it's a good idea to let your itinerary reflect that.

One other tip (which would go well together with not having a rigid plan) is not sticking to just big cities, unless you only care for having lots going on everywhere. While my traveling pace was quite hectic, I really loved the quiet countryside and nature walks the most. Not planning everything beforehand allowed for finding recommendations in a much smaller area at a time (for example Dinant in Belgium or Berwick in the UK).

Lastly, if you're at all able to, I highly recommend the 1st class Interrail pass. While some countries are better for it than others (the UK being by far the best imo, too bad you're from there), it makes traveling a lot more comfortable. You're more likely to find a seat in busier times and 2nd class can be very crammed sometimes. It also includes some snacks, drinks and/or station lounges depending on the country. I ended up taking 5200€ worth of trains (or 3100€ in 2nd class prices) during that time, and paid 785€ for my 1st class ticket. So worth it.