Let me point out that canonical profession is not needed to be a "proper" hermit. We have lay hermits and priests living as hermits --- both without public vows (and often without private vows either). Canonical vows (part of the larger act the Church recognizes as profession) are needed to live and represent eremitical life as a Catholic Hermit, that is, in the name of the Church. If you wish to live as a hermit your bishop can give you permission to do so; strictly speaking you do not need to be professed as a diocesan hermit under c 603. You could, if you desired, make private vows with your bishop as witness (though he would not be "receiving" these vows in the name of the church; that would require profession under c 603). One problem with this option or the next is that in my experience, bishops are generally very reluctant to give permission to diocesan priests to become hermits; not only does the priest shortage make this difficult but the long period of discernment and preparation in one being admitted to the Sacrament of Orders strongly suggests that, short of a life-changing event or circumstances, eremitical life is contrary to the person's true vocation.
Also, I believe as a matter of true governance (and your own responsibility), acceptance of responsibility for your vocation and vows by the second bishop would also require your incardination in the new diocese. What I cannot envision is incardination as priest in one diocese and profession (or reception of one's vows/vocation) and consequent standing as a diocesan hermit in another. In such a case you would be a single subject attempting to live under the canonical authority of two different bishops and that strikes me as incoherent with neither bishop really having true jurisdiction. I doubt a bishop can simply relinquish authority in the way you have described.
Since I am not a canonist, however, I will refer you to one whom I know and trust with particular expertise with canon 603 but also in matters having to do with ordained and consecrated life more generally. While I believe I have given you accurate information, a second opinion might be of assistance. Meanwhile, I hope my response is helpful both as a direct answer to your question and as a way of thinking further about canon 603 vocations. Whether private or public commitment, whichever option you choose, I wish you good luck in your journey to/in eremitical life!
N.B. The canonist mentioned above commented on the submitted question and essentially noted that it was a matter of jurisdiction and that a priest could not be bound in obedience to two different bishops in two different dioceses. Incardination binds a priest in obedience to the local ordinary; so does canon 603. The first bishop has no jurisdiction over affairs in the second diocese and so, cannot act to delegate authority or give permission in the way described in the question --- something I had not thought of at all myself!