That's a great tour you have in mind there. But man, that's a whole load of miles you're looking at. Comfort is not optional: you'll be in that saddle all day most days. You want to enjoy this right?
Western European speeds are high. You gotta keep up. Motorways are everywhere, and often very useful for getting from A to B when the in-between is less than stellar.
600cc is a standard size for multi-purpose biking in western Europe; 125cc standard for about town. There is good reason for this.
A small single bike will be at its limit most of the time on any road of size. Try sitting on a washing machine for an hour with a hairdryer in your ear (in a gale if you can manage that). The thrill fades. Doing those distances though Scandinavia western Europe and on a 125 is demanding. And at speed, not so pleasant. I wouldn't.
A faring for example, will make a huge difference in your enjoyment levels.
For the tour you've described, there are loads of bikes that will deliver what you want - all those Japanese commuter twins for example. Ideal for a beginner (a single wont be particularly economical at standard highway speeds). A 500/600cc multicylinder fared bike will give you both safe levels of speed and decent economy. They are reliable, easily found and cheap. Loads to choose from. You can also find them with built in luggage: far superior to bungee cord mayhem four times a day.
Looking at your money levels: do you really think you can live on a grand a month? That's a tall order -- even excluding fuel from that allowance. You want to crunch those numbers a bit more I think. It implies a serious level of deprivation. Camp sites in western europe are about 20 euros a day during the summer for example.
In France, Italy, Scandinavia I would budget several hundred pounds a week, and that's being moderately careful.
Another option: how about focussing on a smaller area; quality over quantity? I would imagine a backroads tour of a less vast land mass would be great fun on a 125. You could make friends and really get to know an area.
When you scale down in this way, everything becomes less demanding and less expensive: luggage, gear, planning, targets, and so on. If you choose eastern Europe then your budget is going to go a lot further and a small bike better suited to the road conditions.
Apologies for the negatives: I am trying to help. An enjoyable trip needs realistic logistics. They free you to have fun. Couchsurfing though Romania and Ukraine anyone? There are worst things for a 27 year old man to be doing with his days.