Why was Descartes' interpretation of the Catholic transubstantiation not approved by Church and how can it be improved?

Descartes didn't believe in substance—or, what amounts to the same thing, he redefined substance as simply "extension" (length, breadth, width).

Thus, for Descartes,

1. substance = "extension" (length, breadth, width).
2. Catholic doctrine:
   The substance of the Eucharist is the body, blood, blood, soul, and divinity of Christ.
   ^{(cf. Council of Trent session 13 canon I or St. Thomas Aquinas's Summa Theologica III qq. 73-78 et 79-83)}
3. ∴, for Descartes:
   the body, blood, soul, and divinity of Christ = "extension" (length, breadth, width),

which is quite a blasphemous statement! Christ is not just three numbers!

Descartes views on the transubstantiation are thought to be why his works were placed on the Index in 1663 and why King Louis XIV wrote an edict in 1671 prohibiting the teaching of Cartesianism in France.

See ch. 7 "Descartes and the Jesuits of La Flèche: The Eucharist" (pp. 217 ff.) of Descartes Among the Scholastics.

As you describe the issue, it seems to be the case that Descartes was completely rejecting the distinction between substance and accident which is the basis for Catholic (and other) belief in transubstantiation. The Stanford Encyclopedia of Philosophy article on "Substance" bears this out:

Descartes, like the atomists, believed that matter operates in an entirely mechanical way. There is, therefore, no causal role for substantial form to play and, hence, no need for such forms. His two substances are each defined in terms of one property (extension for matter and thought for mind), hence there is no problem about the relation between substance and the properties in terms of which it is defined. As he does not have substances as individuals made of kinds of stuff, there is no conflict between individuals and stuffs.

That is, Descartes did believe that there was a substance that was "what something was made of", but he didn't believe (as for instance Aristotle and Aquinas had) that there were different kinds of this substance that made different things what they were. Loosely speaking, Aquinas believed that an apple was not a chair because an apple had "apple-ness" (the substance that made it an apple) and a chair had "chair-ness", whereas Descartes believed that an apple was not a chair because an apple was "apple-y" (it had properties, or accidents, peculiar to apples) and a chair was "chair-y".

But believing this, the Church thought, left you unable to believe that the consecrated Host was really Jesus. Surely all the properties that made Jesus "Jesus-y" had vanished with his body; how could the "Host-y" properties become them? Or if the Host's properties (size, weight, taste, smell, etc.) vanished and were replaced with Jesus', how could this happen outside of some sort of illusion or hallucinatory experience?

Some theologians in the seventeenth and eighteenth centuries, who inclined to Cartesianism, as E, Maignan, Drouin, and Vitasse, displayed but little theological penetration when they asserted that the Eucharistic appearances were optical illusions, phantasmagoria, and make-believe accidents, ascribing to Divine omnipotence an immediate influence upon the five senses, whereby a mere subjective impression of what seemed to be the accidents of bread and wine was created. Since Descartes ... places the essence of corporeal substance in its actual extension and recognizes only modal accidents metaphysically united to their substance, it is clear, according to his theory, that together with the conversion of the substance of bread and wine, the accidents must also be converted and thereby made to disappear. If the eye nevertheless seems to behold bread and wine, this is to be attributed to an optical illusion alone. ... Both philosophical and theological arguments were also advanced against the Cartesians, as, for instance, the infallible testimony of the senses, ... the striking expression "breaking of bread" (fractio panis), which supposes the divisible reality of the accidents, etc. For all these reasons, theologians consider the physical reality of the accidents as an incontrovertible truth, which cannot without temerity be called in question.

(New Catholic Encyclopedia, "The Real Presence of Christ in the Eucharist")

Because it appears impossible, then, for Descartes' philosophy to simultaneously say that the "accidents" (the properties) of the bread all remain, while the entity itself becomes truly the Lord Jesus, it is similarly impossible for the Catholic Church to accept it as a description of what goes on at the Consecration.

I can't see any way of adapting this Cartesian explanation of transubstantiation without completely changing the Cartesian understanding of "substance" and "property"/"accident"; so I have to conclude there's no way of improving his theory to make it acceptable to the Church.