According to Wikipedia Ivan Illich opposed "institutionalized education" or "compulsory mass education", but he favored "self-directed education, supported by intentional social relations, in fluid informal arrangements".
It may be a misrepresentation of his position to simply say he was "so opposed to schools" unless one clarifies that what he was opposed to was an institutionalization of society one example of which was mandatory public education. Another example of this institutionalization that he did not favor could be found in his book Medical Nemesis.
He favored "learning webs". His language school, training for missionaries to Latin America and sessions of seminars at Cuernavaca in Mexico (Centro Intercultural de Documentatión) may have been an attempt to illustrate how a de-institutionalization could be accomplished. One might look at libraries and these stack exchanges as learning webs.
The OP asks this question:
How did Illich reconcile his love of the Catholic Church with its many schools when he himself was so opposed to schools?
Having schools is not the problem. Rather the problem is institutionalized, compulsory mass education. So his position need not be directed against the schools run by organizations such as the Catholic Church as long as they did not encourage an institutionalization of society.
Wikipedia contributors. (2019, January 21). Ivan Illich. In Wikipedia, The Free Encyclopedia. Retrieved 22:10, March 11, 2019, from https://en.wikipedia.org/w/index.php?title=Ivan_Illich&oldid=879396399