In terms of the biological mechanisms, you have two major mechanisms that come into play:
The first is the random segregation of chromosomes during meiosis, the generation of the egg cell. The egg cell will contain only one of each chromosome, and you need it to be the "maternal" chromosome each time. This is unlikely (1/2^23 for each generation) but unlikely is not a problem.
The second is the crossover mechanism that occurs during meiosis. The parental chromosomes are not directly passed on, they mix with each other and this means you get genetic recombination. So chromosome 1 in an egg cell isn't the same as the maternal chromosome 1 of the individual, or the paternal chromosome 1. It's a shuffled mix of both maternal and paternal chromosomes.
The exceptions to this rule are chromosome Y which is passed on directly from father to son and the mitochondrial chromosome which is passed on directly from mother to daughter.
It's estimated that crossover occurs around 75 times for human females (55 for human males), but there is nothing biologically that seems force the crossover event, aside from overwhelmingly high random chance. While there doesn't appear to be any mechanism that would check that a crossover has occured in order to determine if the meiosis is allowed to occur, it's extremely likely that it is physically impossible for the chromosomes to all be present during meiosis without enduring crossover events. Not everything is known about how crossover events are regulated but the systems seems to be rigged to always have crossover events occur during meiosis.
This makes it a little trickier to argue that it happened despite being unlikely because it might really defy fundamental physical forces. 29
That being said, ignoring the crossover problem and suggesting that it is simply forced random chance seems reasonable enough because there are no known "checks" that crossover has in fact occurred that would stop a cell from completing meiosis if it hasn't.
If you want a stronger case, I would suggest explaining that the mother passes on 1/2 of her genes and her entire mitochondrial DNA to M2, F1 passes on 1/2 of his genes to M2. Now M2 has half of your genes and half of F1's genes.
Thinking about it a little deeper, I think that the problem you try to solve is not the one you outlined. You're asking if it's possible to "replace" your own grandmother, but you frame the problem as needing to provide 100% of your own genes.
M2 and F2 each pass on half of their genes to you. So you regain 1/4 of your original genes from M2. But you also have 1/2 of F2's genes and 1/4 of F1's genes.
Why can't the 1/4 of F1's genes happen to correspond to exactly 1/4 of your genes, and the 1/2 of F2's genes happen to correspond to exactly 1/2 of your genes?
To think about it a different way, you already know that your maternal grandparents, together, had 1/2 of your genes. You also know that your father has the other half. By going back in time and becoming your own grandmother you're only changing the maternal 1/4, but you know that this 1/4 already existed.
The case in which you need to provide 100% of your own genes is if you go back in time and, instead of replacing your own grandmother by having a child with your grandfather, you try to become your own grandmother by having a child with someone who is not genetically related to you in the original timeline.
The interesting thing about your question, though, is that it's only possible if a girl replaces her maternal grandmother. Replacing your paternal grandmother breaks the direct heritage of the mitochondrial genome. Conversely, a boy cannot become his maternal grandfather. 882