Thank you for writing. Assuming the situation (the illness) is a serious matter, I have to say frankly that I think your specific idea is really terrible. While I understand the fear you are experiencing, it makes no sense to approach your diocese with a petition to admit you to eremitical profession while considering withholding important (in this case critical) personal information from them. While not every form of illness needs to be disclosed to the diocese professing you, truly serious illnesses that impact the way you write and live your Rule do need to be disclosed to and understood by your diocese before they agree to profess you. (Not least, any form of chronic illness must be considered and assessed as the diocese discerns one's ability to live the life one proposes to live in the name of the Church. This includes mental illnesses, and certain neurological illnesses or disorders that are progressive in nature, or which are intractable.) To refuse to do this would be tantamount to a lie. Canonically, I believe the Church could determine your profession to be invalid in such circumstances (they could be said to involve fraud), but, as I am not a canonist, I would need to check that out.
Canonical matters aside please consider the wisdom and import of approaching public profession while withholding such a significant piece of personal information. If it is serious, your chronic illness is not something peripheral to your life, whether as a hermit or not, but central to it and to the witness you are called to give to the Gospel. Is there a dimension of your life and identity that is not touched by your illness and its requirements? In light of this, how will you write a Rule of life that binds you in law if you do not include the fact of chronic illness? How will you be bound in obedience to legitimate superiors who do not know this important truth about you? (In this matter consider how they would exercise a ministry of authority --- which is a ministry of love --- if they know you so incompletely or partially and in such a significant matter.)
Finally, please consider that many diocesan hermits have chronic illnesses while others are aging and becoming more or less disabled in this way. We are finding our way in this as in many things. In my experience, dioceses do not usually refuse to profess a person simply because of a chronic illness if that person can live the central elements and spirit of eremitical life at the same time. Some illnesses will not allow this (nor will some vocations), but since a major part of eremitical solitude is its distinction from isolation, most of us find that chronic illness is something eremitical life can redeem in ways that allow illness to be a significant witness to the individual's true value even (and maybe especially) when eremitical life does not occasion healing from the illness itself. If one cannot risk being truthful in this matter it may suggest that one is simply not suited to the risk of eremitical life itself or the radical honesty it demands --- at least not at this point in time. On the other hand, if one's diocese is talking about making a blanket rejection of chronically ill hermits, perhaps it is time for candidates to educate them, at least generally, regarding the place of chronically ill hermits in c 603 vocations.
To educate one's diocese in this way, however, means you must live the truth in a transparent way, and doing so long and faithfully enough that you can articulate it clearly for your diocese. Eremitical life itself is edifying; the eremitical life of one who is chronically ill or disabled is meant to be doubly so because it demonstrates what is possible when God is with us in abject human poverty. The basic question your own query raises and which one must answer convincingly will always be, which does one desire more, to live eremitical life and serve the merciful God of truth in this way or to be professed canonically? Canonical profession can and does serve our living out of eremitical life, especially as an ecclesial vocation, but it is a means to the journey of radical truthfulness, authentic selfhood and holiness; it is not the end in itself. You would betray all of that if you had a lie or serious deception at the heart of your profession.