How can I best address this plan of mine with my manager?
A private meeting, phone call or video chat. I wouldn't say "girlfriend" to him. Simply "family" or fiance/wife (if you had that option) carry much more weight, at least in my experience but maybe the culture is different on that issue where you are. I imagine that his reaction will be related to how often he thinks in person meetings will be required and how much advanced notice he thinks he'd be able to give you. You could ask him about the frequency of meetings and the lead time first.
An example question for the boss: "During the busiest time you can recall, how often did someone at my level and 1 level above me have to come to an in person meeting? And how far in advance did they have to schedule these?" Now this question hints at the other part of this, your role/seniority matter too. The frequency probably increases the higher up you go and ideally you're going to get promoted in the next 1-3 years. Your co-workers who are at higher levels can probably give you a better answer too since they will know the frequency of in person meetings with other departments. Your boss might not know that since if it's like anywhere I've worked, he's only interested in progress/results/issues. Not the details of where/how you met with others.
Typically if you're meeting with other departments there's a little more flexibility in scheduling than when your boss wants to meet with you so you might be able to get 2 meetings to occur on the same day instead of spread out over multiple days but it's unpredictable since the schedule of the people you're meeting with might be too packed for them to accommodate your needs. My other observation is that when you're a lower level person meeting with a high level person in another department, they tend not to like it if you're even 1 minute late for a meeting. If your mode of transportation has too much unpredictability in it you have to account for that. Now your 7 hour trip might actually take 8-12 depending on how early you need to leave to eliminate the risk of travel delays.
Is this something that can be done at all or am I just fantasizing and there is no way this is practicable?
I think you're fantasizing - not because it's impossible to get the change you want - because if you were to prioritize your life goals, I suspect that starting your career with all of the investment you've already made in this company is a higher priority than living with your girlfriend. Likewise it is a higher priority to her for her to finish her apprenticeship than it is for her to move to city "B" and be near you should you live 7 hours away from her school.
The risk of living in a city that is 7 hours travel distance from where you may need to make an in person appearance for business meetings on short notice is huge. Imagine if the boss calls a morning meeting the evening before. You would then need to start travel within the next 3-4 hours or beg the boss to excuse you. Imagine if there are multiple meetings in the same week called by people in different departments. Your boss cannot guarantee that this won't happen. When you eventually work on projects that require you to deal with management level people in other departments it's outside of your boss's hands. Or imagine if your boss transfers to another department, gets promoted or otherwise leaves the scene.
To give yourself the best chance of success you need to be near your office. I have known people who lived just 2 hours away from the company HQ and despite being department management level people working remotely, when the company had to reduce costs they were terminated long before lower level people working remotely in the local area.