There is no universally standard name for that exact metric. The cleanest term would be:
knee-optimal, utility-weighted F_\beta score
or more compactly:
utility-calibrated F_\beta at the Pareto knee
The pieces mean:
F_\beta: weights false negatives versus false positives through \beta.
Utility-calibrated: choose \beta from the relative marginal cost of FN and FP, rather than arbitrarily worshipping F_1, humanity’s favorite symmetric default.
Knee-optimal / Pareto knee: choose the operating point where further FN reduction requires sharply increasing FP, the point of diminishing marginal utility.
A mathematically explicit label could be:
F_{\beta^*}@\text{knee}
where \beta^* is determined from your FN/FP utility ratio and the threshold is chosen at the knee of the precision–recall, ROC, or utility frontier.
One important distinction: a knee point is not necessarily an inflection point. An inflection point is where curvature changes sign; a knee is where the marginal tradeoff deteriorates most sharply. Your description sounds more like a utility knee than a strict inflection point.
For a paper, I’d write:
Utility-calibrated knee-optimal F_\beta
and define it once as “the maximum utility-weighted F_\beta attained at the knee of the FN–FP tradeoff frontier.”