Carvaka as sarvadarsana pratikakshi
Rejection of vedic authority carvaka's may be classified with buddhist,jaina and other shramana traditions. But in their world-view there is a difference while buddhist and jaina traditions are pessimistic about this world and argue for renouncement - the debate b/w vyasa and his son in mahabharatha may be expressing this debate and ashrama system can be considered as a reconcilation of two views. The belief in rebirth , karma and moksa can be considered the basic common beliefs of both vedika and avaidika traditions even-while they vehemently disagreed on details like nature of self, working of karma and state of liberation. Carvaka is an outlier so may be they are called "sarvadarsana pratikakshi".
Carvaka as Optimistic Buddhist - If we focus on Jayarasi a sceptic symphathetic to carvaka position in his work Tattvôpaplava-siṁha (‘The Lion of the Dissolution of [all] Categories’) he is sceptical of philosophy i.e. epistemology itself arguing we can't know much and these speculations are distractions from enjoying life he concldues “When, in this way, the principles are entirely destroyed, all everyday practices are made delightful, because they are not deliberated”. One may compare to Buddha's mahamouna and his dismissing of metaphysical speculation saying it distracts from the immediate aim of cessation of dukha.Both seems to represent pragmatist bent of problem solving and criticises metaphysical speculation as distraction. But their radically disagree on nature of world for buddha it is full of suffering and he gives a image of a person struck by posinous arrow so he should be treated instead of speculation on its who shooted it etc.May be jayarasi will give a image of tasty food one should eat it instead of indulging in unneccary speculation.
Carvaka as Mimamsaka who lacks belief in vedas
I agree both Carvaka and Purva-Mimamsa are both Darsanas, i am not demeaning the carvaka view-point but reflecting on the two darsanas.
Mimamsaka's were accused of half-carvakas (ardha-lokayatas) whcih kumarila famously wanted to refute it. Purushottama Bilimoria in his essay "Hindu Doubts About God: Towards a Mimamsa Deconstruction" discusses it in detail and explains kumarila's main problem with that characterization is the ethical implication of hedonism not the ontological question of materialism. So may be we can call a mimamsaka as "realist who vouches by the vedas" - representing both its realism and belief in the authority of the vedas.
So by saying "carvaka is a mimasaka who lost faith in the vedas"I was reformulating the statement that mimasaka is ardha-lokayatha.
From the mimamsa position if one takes away the belief in vedas and associated notion of karma,apurva etc the left out theory will not sound like caraka position?Qouting from [https://plato.stanford.edu/entries/naturalism-india/]
//The Mīmāṃsaka-s admit the doctrine of karma but supports non-naturalism in moral context. They believe in two causal realms — ritual and natural and according to them the ritual order is independent of the natural order. In fact the causal connection that obtains between a ritual enjoined by the scripture and its result cannot be explained naturally, e.g., how the correct performance of putreśṭi sacrifice fulfills one’s desire for a son is beyond natural and scientific explanation//
The ritual world may stand for his belief in vedas, so carvaka can be said to believe in only one natural world.
This is further supported by the this-wordliness as in positive attitude about loukika world compared to complete renouncment of shramana traditions like Jainism and buddhism.
Considering the historical development of Purva mimamsaka throuh the works of jaimini sutras and later sabara,kumarila,prabhakar may be considered as mature articulation of ideas in karma kanda of vedas empahasing yagna and rituals i would speculate there was a sceptical stream of thinkers best expressed in nasadiya sukata whose ideas may have given rise to carvaka position as they kept the this worldly focus but rejected the authorith of vedas and efficacy of rituals mainly due to scpetical attitude towards sabda and anumana pramana.