Tldr: Yes, it may be more cost-effective to prevent than to treat obstetric fistulas. Yes, there is an organisation working on preventing fistulas, and yes, TLYCS should probably consider it, but not necessarily for that reason. The cost of preventing vs treating obstetric fistula (OF) is comparable, but preventing OF has additional benefits such as preventing the associated stillbirth, infertility, psychological trauma, and social detriments. Furthermore, the life-saving value of training midwives in low HDI countries far outweighs the value of OF prevention generated by the same intervention. 

Fully training one midwife to international standards, through the Catherine Hamlin Fistula Foundation (CHFF) costs 18,000 AUD. I would estimate that one midwife is likely to attend something closer to 5000 births across their career, although they may also drive change by providing family planning services and directly or indirectly educating many members of a community besides the birthing women they attend. 

According to CHFF, 

"When a Hamlin midwife is placed in a rural area [in Ethiopia], the number of new fistulas drops to zero in nearby villages!" 

We know that OF is highly preventable with basic educational and healthcare shifts because it is virtually unheard of in high-income countries, even in comparatively disadvantaged populations. If "drops to zero" reflects an OF RR of ~0-0.2 in communities cared for by CHFF midwives, the prevalence of OF in these areas is ~0.4% per birth, and one midwife attends ~5000 births, that midwife may only prevent 16-20 obstetric fistulae across their career. That means simply training midwives in Ethiopia (and comparable countries) could prevent OF in the absence of any other interventions, for a similar cost per fistula to $619 surgical repair, but with far greater reduction in suffering. In particular, prevention of OF also prevents many of the stillbirths which would otherwise occur alongside 93% of cases. Similarly, OF is strongly associated with infertility; divorce; mental health conditions; and years of a mother's separation from her children and community, due to the stigma of OF-related incontinence. These flow on effects of OF are far more difficult to reverse with surgery than the injury itself. 

To me, it seems unlikely that additional interventions, with the possible exception of travel stipends, would increase the cost-effectiveness of prevention, compared to midwifery training alone. I am assuming that midwives can provide key education, e.g., regarding child marriage, early pregnancy, and nutrition, and that facilities for common obstetric interventions are already available. It seems likely that available medical resources in areas with high rates of OF are under utilised, and would continue to be so even with greater funding, due to ~70% of birthing women lacking an attendant trained in detecting and escalating intrapartum abnormalities as needed. If these assumptions are valid, investing in media interventions, training doctors or building hospitals/specialist clinics would provide negligible additional benefits. 

However, it may be possible to modify the midwifery training approach to prevention to increase cost-effectiveness. For example, up-skilling traditional birth workers to safely care for low-risk birthing women, including recognising and escalating common complications, could provide many of the benefits of fully-trained midwives at a lower cost and higher cultural acceptability. 

All that being said, while training midwives (or even traditional birth workers) to prevent OF may be only marginally more cost-effective than curative healthcare, the "side-effects" of this intervention are far more impressive. The primary benefit of training midwives in low HDI countries would not be OF prevention, but stillbirth and maternal and neonatal mortality reduction. According to Nove et al. (2021), 

“Achieving a substantial scale-up of coverage of essential interventions that can be delivered by midwives who are educated, regulated to global standards, and working within an enabling environment by 2035 could avert 40% of maternal and neonatal deaths and 26% of stillbirths, relative to those projected to occur under current coverage. Achieving universal coverage could avert 65% of all these deaths.”

Ethiopia, where CHFF is based currently has baseline rates of 0.267% maternal deaths (mat. D) and 4.18% early neonatal deaths (neonat. D) per live birth (LB) and 0.92% stillbirths (SB) per birth (B).

$\frac{({0.00267}_{\text{mat. D}}+{0.0418}_{\text{neonat. D}}){991}_{\text{LB}}+{9.20}_{\text{SB}}}{{1000}_{\text{total B}}}\ast {5000}_{\text{midwife B}}\ast {0.65}_{\text{of D and SB averted}}={173}_{\text{lives saved per midwife}}$ 

$\frac{\${18000}_{\text{per midwife}}}{{173}_{\text{lives saved per midwife}}}=\${103}_{\text{per life saved}}$

I'm sure there are many other complicating factors (or maybe my maths needs to be corrected?), but to me, that seems like a similarly, perhaps even more, surprising and exciting estimate than $20 to prevent obstetric fistula!

The WHO's article on midwifery education and care has some further information about the global health benefits of midwifery training which I haven't mentioned here but may be of interest.