Must admit I am also surprised and dismayed at the vitriol directed at the OP. And the disrespect for the girls' stated boundaries.
Yes, there are some 12 year-old girls for whom tampons are an option and yes athletes intent on competing will do whatever it takes to minimise disruption. Taking the pill is not just one way to do this, it is actually one of the main ways menstruation can be managed (by for instance taking the pill in a way that stops periods altogether).
However, as Jessica says above, messing with the endocrine system of grown women is not without risk, and that risk is much larger for preteen and early teen girls.
As for whether or not this is a feminist issue. This is a booking system based on the default: the average healthy male. If default males had regularly occurring downtimes when they could not attend a monthly session, you bet your bottom dollar flexible schedules such as described by PP would be the norm.
Systemic oppression and structural discrimination are also expressed in standards that do not ever take the needs of a particular group into account.
Take for instance hand dryers in public toilets, automatic towel rolls, soap dispensers or taps. Many of those use sensors that are blind to anything other than the standard Caucasian skin tones. Thus, many POC find that these devices do not work for them. Were they designed to prevent POC from using them? Of course not. But the designers did not take them into account because their default customer is white and that's who they designed these for.
A system which disadvantages entire groups of people is a system of inequality.
That's what we have in this schedule. Because the default is also the healthy male, these scheduling and booking conventions also discriminate against swimmers with disabilities who may miss sessions regularly due to their health, such as needing to attend monthly clinic appointments for instance.
Acknowledging that systems based on the average healthy male may not suit the needs of other groups is neither ridiculous nor does it make a mockery of feminist issues.
This is the feminist issue - that we live in a system geared towards meeting the needs of males. Designed by and for males, governed by and for males and regulated by and for males.
Menstruation is a regularly occurring biological function of the average healthy female. If we didn't live in a patriarchy, swimming schedules would take this into account. And by now we would have found an equitable solution that met the needs of customer and provider. They do exist as we can see from a number of suggestions in this thread. Some will be financially more viable than others. No equitable solution will require young girls to insert a foreign object into their body when this is not what they wish or to medicalise them unnecessarily.
As my mother always says, where there is a will, there is a way. The OP has come here to ask whether there is a way to address an issue that only girls and women qua females can ever face - that of losing out financially because of a standard biological function of our bodies.
Is it a huge issue? On a par with the worst transgressions against females? No. But systemic oppression and structural discrimination are made up of a multitude of factors, many of which are small issues like this one. This is a feminist board, not a huge-issues-only-feminist board, so I do think it is a discussion that belongs in this space.