Suppose one player has only 3 minutes of time left in his clock. The position is a draw with opposite colored bishops and neither player can break forward practically. The other player, who has got lots of time left, intentionally plays on to win on time. Since the player under time pressure will not be keeping score, how can he claim if he has completed 50 moves as per the 50 move rule? What is the best solution to this situation in a tournament organized according to FIDE rules?
According to clause 8.5a of the FIDE Laws of Chess (the URL keeps changing so I won't link to it, but you can search), if a player has under five minutes left and the increment is less than 30 seconds per move (or nonexistent), an arbiter or assistant should keep score if possible. This is one reason that a 30-second increment is used these days; both players are required to keep score for the whole game so this situation never comes up.
If the game is played by USCF rules, then according to rule 14F4, if a player has under than five minutes left, he can ask the tournament director to keep track of the number of moves in order to verify a claim that the player makes. (The player is still responsible for making the claim.)
The FIDE rules have a solution for you. If you have a 30 second increment then you have to write the moves down in any case. If you don't get the increment then the Quickplay Finishes section of the rules in Appendix G apply and article G.5 fits your problem. Note that this applies only to Standard and Rapid games. In Blitz games you are on your own. Losing on time in such a position is part of the game.
Note that article G.4, which is referenced, just says that with less than 2 minutes left you can request a 5 second increment if possible. You would only do this if you were trying to win and needed more time.
G.5 If Article G.4 does not apply and the player having the move has less than two minutes left on his clock, he may claim a draw before his flag falls. He shall summon the arbiter and may stop the chessclock (see Article 6.12 b). He may claim on the basis that his opponent cannot win by normal means, and/or that his opponent has been making no effort to win by normal means
What this means is that you have to continue playing until you have less than 2 minutes left on the clock, then stop the clocks and call the arbiter and make your claim. In your situation the arbiter will defer his decision, ask you to play on and watch what happens. He will then decide if in his opinion your opponent cannot win by normal means or is making no effort.
If he decides in your favour you get a draw. If he decides against he will give your opponent an extra 2 minutes on the clock.