I love this question. Because it's not just a hypothetical. We do have to choose.
Everyone can say who they'd select -- but if we're considering who Fox (&Pace) would select, I think some are still amusingly considering FoxPace wants to lose games for a better draft pick (which they aren't considering), or that winning in 2016 isn't a top priority (it is). These jobs are hotseat.
To Win More 2016 Games: Fuller
The health of Fuller and White is going to play the major factor, but if they were both ready to go, without a reasonable concern that this return would cause re-injury, then I think there's no question the team needs Fuller way more than White. Meredith & company is more sufficient than our CBs without Fuller (Callahan, Hall, Tracy injured).
Assessing for 2017: White
We don't need to see more of Fuller. We know what we have with him.
We haven't seen enough of White yet. If White comes back, and lights it up, Bears don't have to overpay Jeffery. We have a need that Jeffery can fill, but not at all-costs. If White is perfectly healthy but is hapless, then our need for Jeffery increases.
Good Guess: You just have to watch White's last college year to see he has a lot more skills than speed, along with excellent size, strength & jumping ability. If he can stay healthy, he'll produce. His bones are the concern, not his ability.