If you have a Mass or Office for him on July 9, I'd pick that date. There's plenty of saints whose feasts are moved because their dates of death are occupied.
The patron of your diocese is a Double of the I Class, so it would take precedence over any lesser feast on that day, with the other feast commemorated, transferred, or omitted, according to the Rubrics. If, however, another feast would perpetually take precedence over the patron's feast, it would be transferred to the next day which does not exclude Doubles of the I Class.
Lower-ranking feasts particular to some nation, diocese, order, institute, or church, if perpetually impeded in the entire nation, diocese, order, institute, or church, are moved to the next free day. If only impeded in certain places but not others, the feast is commemorated or omitted, according to the Rubrics.
If your question is slightly different - if the Mass and Office is for two saints (Ss. John Fisher and Thomas More, or Ss. Perpetua and Felicity, or Ss. Simon and Jude), where only one is the patron, the feast of both is kept as usual, but as a Double of the I Class, unless there's a proper Mass and Office for that diocese with only the one saint.
Title V. On the perpetual occurrence of Feasts and their reposition
1. Feasts of Double rite whether greater or lesser or Semidouble of the universal Church, whether fixed or mobile, if anywhere they are perpetually impeded, are not replaced, but commemorated or omitted on its day, as is said in Title VII On Commemorations. Likewise it is kept of Feasts of some Nation, or Diocese, or Order, or Institute, which equally, if in some particular Church they are impeded on their day, are commemorated or impeded, as above. But proper Feasts of some Nation, Diocese, Order, Institute, or particular Church, which in the entire Nation, Diocese, Order, Institute, or particular Church are respectively impeded, are replaced on the nearest following free day, according to the Rubrics.
2. Fixed Double Feasts of the I and II class perpetually impeded are replaced, just as in their proper place, on the first day free from another Double Feast of the I or II Class, from a privileged Vigil, and from other Offices which respectively exclude this sort of Feast. But Doubles of the I or II class assigned to certain Ferias, if perpetually impeded, likewise are replaced just as in the proper place on the next following Feria perpetually free, as above.
3. Sundays, whether greater or lesser, exclude the perpetual assignment of any Feast even a Double of the I Class, excepting the Sunday between the Circumcision of the Lord and the Epiphany, on which is said the Office of the Most Holy Name of Jesus, the Sunday within the Octave of the Epiphany, on which is said the Feast of the Holy Family of Jesus, Mary, Joseph, the I Sunday after Pentecost, on which is celebrated the Feast of the Most Holy Trinity, and the Sunday before the Kalends of November, on which is recalled the Feast of O. L. Jesus Christ the King.
4. The Commemoration of All the Faithful Departed excludes not only occurring Feasts but also transferred Feasts of whatever rite.
5. If within an Octave or on the Octave day itself, some Feast of the same person perpetually occurs, all is observed which was said on accidental occurrence in title IV, num. 7; but excepting the Feast of the Holy Family of Jesus, Mary, Joseph, which is ruled by special Rubrics set out in the Proper of the Season. 6. When Feasts of the universal Church which are celebrated with Octaves, because of a perpetual impediment, are to be replaced on the next following day, the Octave day is not therefore replaced, which ought to be recalled on its day in the universal Church. The same is to be said of an Octave day of a proper Feast of some Nation, Diocese, Order, or Institute, which in some particular Church, are to be replaced on another day. On the contrary, if a proper Feast of some Nation, Diocese, Order, Institute, or particular Church which are celebrated with an Octave, are impeded in the whole Nation, Diocese, Order, or Institute , or in its particular Church respectively, and therefore it is replaced according to the Rubrics, the Octave day is also replaced, which is to be celebrated on the eighth day after the celebrated Feast, as if the same is recalled on its proper day.