There is room for the Likud and Bayit Yehudi to back down. However they are pressing for penal sanctions (as opposed to cutting off subsidies), and that means a de facto war with the hareidim. Likud and Bayit Yehudi could stop demanding criminal sanctions for those refusing to serve in the army, but they have certainly not been acting that way. Labor could offer to cancel the penal sanctions, and make long lasting ally with the hareidi parties (much to the detriment of Likud and Bayit Yehudi).
It should be noted that many on the left also don’t want to serve in the army, so Herzog can please two groups by ending conscription (or allow ing exemption for any religious or moral reasons), and that in economics and increasingly matters of national security, the hareidim are “left” to begin with (especially since the hawkish pro-zionist hareidi were stabbed in the back by their former allies in the Dati Leumi/Settlements camp).