I can't offer analysis of your hypothetical ideal case, but in real cases of small or random objects in the solar system, it seems that the trajectory becomes sufficiently indeterminate with increasing intervals from the present, that collisions and other events become really unpredictable. The most likely next event is usually a somewhat close but non-collisionary encounter that perturbs the orbit enough and uncertainly enough to render the subsequent course unpredictable.
For example 2060 Chiron, a minor planet and/or giant comet discovered in 1977 with a period presently of roughly 50 years, has been studied by both forwards and backwards integrations of its orbit -- see the report "Rapid dynamical evolution of giant comet Chiron", G. Hahn & M. E. Bailey, Nature 348 (1990) 132–136 and a study by Horner et al. (2004).
The indications for Chiron are that it has had and will again have perturbing encounters, especially with Saturn, not very close to actual collision, but enough to change its orbital axis and period. At intervals beyond such events, the course is increasingly uncertain. In effect, the closer any perturbing encounter, the more the departing trajectory and its estimation are affected by uncertainties both of real trajectory, due to arbitrary tiny influences, and of estimation, caused by any errors in the account. So the medium-long-term future of Chiron and other comparable objects can only be described in terms of statistical chance. Hahn and Bailey reported: "Simulations extending ±100,000 years from the present suggest that on this time-scale, Chiron is about twice as likely to have been a short-period comet at some time in the past as to become one in the future. The mean half-life for such evolution is ∼0.2 Myr, much less than the ∼l-Myr lifetime for ejection from the Solar System, implying that Chiron may have been a short-period comet in the past, and will probably become one in the future."
So the conclusion seems to be that in real cases, possible collisions and other orbital changes can't be identified with any certainty -- unless they happen to be imminent along the present orbital track and due to occur soon enough that there will no appreciable non-collisionary perturbing encounters to cause uncertainties in the meantime.