In Plautus (num)quid causae is found three times (Pseudolus 533, Rudens 758, Trinummus 1188), every time with est.
There may be earlier attestations, but Plautus is pretty old and thus the construction is not a late invention.
I would rather analyze the idiom as quid causae est, including the verb.
Leaving est out is a common feature and would be no surprise.
In the first instance you quote there is indeed an explicit est (now as fuerit).
In the second example there is another verb, exsistere, which is pretty close to esse in meaning.
These with the three examples form Plautus suggests that the verb is a part of the phrase and it might be wise to understand an implicit est with quid causae when no explicit verb is supplied.
Quid and genitive
I suggest taking a look at the Lewis & Short entry on quis, especially section II on the neuter quid.
Section II.A.2 gives several examples of quid together with a partitive genitive, the simplest being quid mulieris uxorem habes?, "what sort of a woman have you for a wife?".
Partitive genitive is common in many uses, and using it together with the neuter quid is a particular idiom.
Such use is not unique to causa, and the ample attestations indicate that the case in question is indeed genitive, not dative.
With this in mind, you could interpret quid causae [est] as "what kind of reason is there", which means "why".
But bear in mind that quid is quite a bit more flexible than the English "what".
It can stand for "why" alone.
The construction quid causae est is a little confusing so it may be simplest to treat it as a fixed idiom in practice.
It could be translated as "what is the reason" or simply "why", depending on context.