When the feast was established in 1264, the feast of Pentecost had an octave1 and effectively lasted a week, up to Trinity Sunday. This meant that the first "unimpeded" Thursday following Maundy Thursday was the Thursday after Trinity — the season of Easter impeded2 all the Thursdays up to Pentecost, which had its own octave covering the Thursday after that.
The text of the Bull Transiturus which established the celebration is published in a Google Book, and it does set the day:
§2 Nos itaque ad corroborationem & exaltationem Catliolicæ fidei, digne, ac rationabiliter duximus statuendum ut de tanto Sacramento præter quotidianam memoriam, quam de ipso facit Ecclesia, solemnior & specialior annuatim memoria celebretur, certum ad hoc designantes, & describentes diem, videlicet feriam quintam proximam post Octavam Pentecostes, ut in ipsa quinta feria devotæ turbæ fidelium propter hoc ad Ecclesias affectuosæ concurrant & tam Clerici, quam Populi gaudentes, in cantica laudum surgant. ...
My Latin isn't really up to it, but I can see that he specifies "the fifth day next after the octave of Pentecost, that is the fifth day of the week" which is the Thursday after Trinity (Sunday is the first day of the week).
This is my attempt at a translation:
And so, we, for the affirmation and exaltation of the Catholic faith, have properly and reasonably determined that besides the daily memorial of so great a sacrament, such as the Church does, a solemn and special memorial is to be celebrated annually, specifically designated for this, and describe the day, namely the Thursday next after the octave of Pentecost, that in this the fifth day of the week, for the sake of the happiness of the devout multitudes of the faithful, clergy with the rejoicing people come together in the churches and arise in a song of praise. ...
A better translation would be greatly appreciated. The last clause at least is a bit iffy.
1 Octave: A period of extending a feast-day over the following week, making eight days in total. See the Wikipedia article. Pentecost is slightly unusual as its Octave does not include the following Sunday. That Sunday is a feast of its own, Trinity Sunday.
2 Impede: A celebration of lower rank is impeded (or suppressed) by a celebration of higher rank. Although Corpus Christi was intended to be "solemn and special", nothing outranks the days of the season of Easter. The General Norms for the Liturgical Year and Calendar say "The fifty days from Easter Sunday to Pentecost are celebrated in joyful exultation as one feast day, or better as one 'great Sunday'", so it would be improper to introduce another element into that celebration. [The ranks of various feasts and seasons during the 13th century are now rather obscure, but Easter has always ranked above everything else.]