But does it also prohibit that I let my clock run down if necessary in the second game (I'm a lightning fast player, one hour would still be a correspondence game to me :-), end the first and then make my move in the second?
There are three FIDE Laws which are relevant:
6.7.1 The regulations of an event shall specify a default time in advance. If the default time is not specified, then it is zero. Any player who arrives at the chessboard after the default time shall lose the game unless the arbiter decides otherwise.
The ‘playing venue’ is defined as the ‘playing area’, rest rooms, toilets, refreshment area, area set aside for smoking and other places as designated by the arbiter.
The playing area is defined as the place where the games of a competition are played.
Only with the permission of the arbiter can:
a player leave the playing venue,
During play the players are forbidden to use any notes, sources of information or advice, or analyse any game on another chessboard.
Your question addresses the second but not the first and third, which are also relevant.
There is no point in turning up 45 minutes late for the second tournament if you were defaulted after 30 minutes.
Solution: talk to the chief arbiter of the second tournament beforehand, explain what you want to do and ideally get their agreement. Worst case is they disagree and you know you have to get to your seat before the 30 minutes (or whatever the default time is) expires.
Ideally the two tournaments are part of some tournament umbrella and the playing areas for both tournaments are each in some joint playing venue. Again you should try and check with the chief arbiters of both tournaments beforehand.
If that is the case then you can go to the second playing area after the start of play, register your presence with the arbiter and your opponent, explain the situation to your opponent, not make a move, go back to playing area one, complete your game and then return to playing area two and start your game there.
A note on a player arriving after the expiry of the default time and the arbiter's decision. When I have been the arbiter in those situations I try and find out what is going on with the late player usually by asking the organizer to try and contact the player to find out what is going on. If the player is just delayed then I ask the present opponent what they want to do in the case that the player exceeds the default time. If they want the point regardless then I default the late player when the time is up. If they prefer to play a game (could be important for norms and tie-breaks, prize money etc.) then I allow the late player to play.