Mugechinashi: good Edo word. Koshigaya Gozan includes it in his Butsurui shōko 物類称呼 as an Eastern (東國) way to say nasakenashi (which is to say "heartless" or "cruel"; the more common contemporary meanings of "pathetic" or "shameful" were still apparently a few decades away in 1775; the first such citation in the Nihon Kokugo Daijiten is from 1813, in Ukiyodoko).

Gozan even offers an etymology for mugechinashi, including a locus classicus in the Nirvana sūtra:

涅槃經二「佛性者名曰無礙智」とあれは佛性のなきといふ事なるべし

In the Nirvana sūtra, it is written: "Those of Buddha-nature are called mugechi"; therefore [the word] must [originally?] refer to someone without Buddha-nature.

There is a slight problem with this etymology, though: the quoted passage doesn't appear to actually be in the Nirvana sūtra, or at least any easily searchable version online today. Perhaps for this reason, the Nihon Kokugo Daijiten cites the Butsurui shōko in its entry for mugechinai, but leaves out the sutra quotation entirely. Instead, there is a note placing mugechinai in the context of an extended family of /mug-/ words dating from the Kamakura period (mugoi has survived to the present day), without offering any speculation as to the roots of the /mug-/ morpheme itself.

It must be admitted, though, that even if 無礙智 is not defined so neatly in the Nirvana sūtra, it is a Buddhist term of art. It is used with a slightly different spelling in the term 四無碍智, for example, usually translated "four unobstructed knowledges" or "... wisdoms." The four are, roughly, knowledge of the dharma (法無碍智), knowledge of the meaning of the dharma (義無碍智), knowledge of different languages (辞無碍智), and knowledge of how to use the preceding three to preach freely (楽説無碍智). Together these are indeed identified with Buddha-nature (see Kim Young-tae's "Wŏnhyo's Conception of Buddha-nature in the Thematic Essential of the Mahāpariṇirvāṇa-sūtra", p199, for example).

The word 無礙智 also appears on its own. For example, it's in the steak-knives speech Mañjuśrī delivers when he first appears in the Angulimāliya sūtra and Angulimāl[y]a asks him what he means by all this "emptiness, emptiness" business ("空空有何義"). Here's the SAT Daizōkyō Text Database version, translated by me, with English equivalents for technical terms mostly according to Charles Muller's Digital Dictionary of Buddhism:

T0120_.02.0527b07: 諸佛如虚空 虚空無有相
T0120_.02.0527b08: 諸佛如虚空 虚空無生相
T0120_.02.0527b09: 諸佛如虚空 虚空無色相
T0120_.02.0527b10: 法猶如虚空 如來妙法身
T0120_.02.0527b11: 智慧如虚空 如來大智身
T0120_.02.0527b12: 如來無礙智 不執不可觸
T0120_.02.0527b13: 解脱如虚空 虚空無有相
T0120_.02.0527b14: 解脱則如來 空寂無所有
T0120_.02.0527b15: 汝央掘魔羅 云何能了知

The Buddhas are like the void: the void has no mark of existence
The Buddhas are like the void: the void has no mark of arising
The Buddhas are like the void: the void has no mark of form
The dharma is also like the void: the Tathāgata's marvelous dharma body
Wisdom is like the void: the Tathāgata's great wisdom body
The Tathāgata has unimpeded wisdom, is free of attachments, cannot be apprehended
Liberation is like the void: the void has no mark of existence
Liberation is the Tathāgata: empty and non-existent
How could you, Angulimāla, fully understand this?

Given all those other /mug-/ words, it strikes me as extremely unlikely that mugechinashi comes directly from the Buddhist jargon. (Not least because the meaning is incorrect: the connection to kindness or mercy would be secondhand at best, via the general perfection of the Tathāgata.) I suppose that if the word mugechi was known outside the temples, though, it might have acted as a mold into which the /mug-/ morpheme was poured somewhere along the line.