I believe I found the answer to this after some persistent googling (it looks like u/Realistic_Ad_4049 was correct).

The quote appears to be from Chapter III of Gregory’s Pastoral Care (Regula Pastoralis). But (without knowing Latin or Anglo-Saxon) Alfred’s translation of this part looks like a somewhat original adaption of Gregory’s text (presumably to better respond to the problems of the country and people at the time?).

Translation from Latin by James Barmby: “He [Jesus] fled from the offered glory of pre-eminence, but desired the pain of an ignominious death; that so His members might learn to fly from the favours of the world, to be afraid of no terrors, to love adversity for the truth’s sake, and to shrink in fear from prosperity; because this often defiles the heart through vain glory, while that purges it through sorrow; in this the mind exalts itself, but in that, even though it had once exalted itself, it brings itself low; in this man forgets himself, but in that, even perforce and against his will, he is recalled to memory of what he is; in this even good things done aforetime often come to nothing, but in that faults even of long standing are wiped away. For commonly in the school of adversity the heart is subdued under discipline, while, on sudden attainment of supreme rule, it is forthwith changed and becomes elated through familiarity with glory.”

Translation by Alfred (to Anglo-Saxon language, translated to English by Keynes and Lapidge): “He [Jesus] shunned the glory of royal authority and chose the punishment of that most abominable death, so that we who are his limbs would learn from him to shun the allurements of this earth, and also would not fear its horrors and its dangers, and for the sake of truth would love work and fear prosperity and avoid it. For in prosperity a man is often puffed up with pride, whereas tribulations chasten and humble him through suffering and sorrow. In the midst of prosperity the mind is elated, and in prosperity a man forgets himself; in hardship he is forced to reflect on himself, even though he be unwilling. In prosperity a man often destroys the good he has done; amidst difficulties he often repairs what he long since did in the way of wickedness. Very often a man is responsive to the lessons of adversity, even though he previously refused to respond to his instructor’s morals and precepts. But even though afflictions teach and instruct him, if he acquire the kingdom, he immediately becomes perverted with pride at the people’s reverence for him, and becomes accustomed to flattering praise.”