Meanings of remorse which do not imply guilt are very old, and it would appear from the OED are not used, and are marked as 'obsolete'. They centre around meanings 4, 5 and 6. the most recent example cited is from 1715. (See below - I have quoted only the most recent examples in each category.)
†4. Regretful or remorseful remembrance or recollection of something. Cf. remord v. 2a. Obs.
1695 W. Temple Introd. Hist. Eng. (1699) 578 Either the Fame of his Forces..or Remorse of his Duty, prevail'd with Duke Robert to offer again his Submissions.
1715 J. Browne & W. Oldisworth State Tracts II. 244 A young, an airy Spark,..now, without Remorse of Gout or Stone, From Sixty odd, sets up for Twenty-one.
†5. Sorrow, pity, compassion. Obs.In quot. 1692 in pl.: signs of sorrow or pity.
1538 Prymer in Eng. after Vse of Sarum sig. kiij, O God mercyfull, pytifull, & fauourable whiche hauynge remorse on the affliction of thy seruauntes, saydest vnto the aungell, [etc.].
1700 Dryden Chaucer's Palamon & Arcite ii. 345 Curse on th' unpard'ning Prince, whom Tears can draw To no Remorse: who rules by Lions Law.
†6. A matter for regret; a pity. Obs. rare.
1576 Humphrey in J. Strype Ann. Reformation (1709) I. xliii. 431 That it was a remorse to seem, by sundry apparel, to sunder himself from those brethren.
It would therefore seem inappropriate, if there is no guilt, even to be starting with the word remorse and looking for an alternative. From a list of synonyms the only one which seemed remotely appropriate was anguish. I think you need to be searching for something completely different, such as regret or sorrow.