For what it might be worth, considering cost per square foot or other method for improving the site, I'd bet there are many alternative that are less cost than grouting.
This somewhat reminds me of folks worrying about the loads from truck traffic on highway fills. In most of those cases the weight of the trucks has very insignificant effect on settlements. The same probably would go for your RR yard loads. Your weights you might more likely need to think about are those soil and rock weights above the former mine workings. I've experienced measuring the effect of loaded freight cars on culverts under the ties and there is significant spread of load under loaded freight cars and locomotives are even lighter (than the cars I worked with).
Time also is important. How long ago was that mine worked? The more time involved, the less likely there will be future RR yard problems.
What does it cost to re-set track in case it tilts or has a roller coaster vertical profile? You could bury this cost in annual maintenance expenses. RR guys easily can drag a whole long section of track, with ties attached, off an area, re-grade it and drag it back. That would be in a real bad case of settlement only.
If you are still concerned and want to do something, consider running a rolling surcharge of earth across the site. I'd think about a "windrow" maybe 10 feet high and 20 feet wide on the top (any length) being shoved across the site with a dozer or a backhoe sitting on top. You could monitor the effect with simple settlement platforms. Moving fill this way is cheap, probably much cheaper than the grouting option. It would put much more load (pressure) on that mine than any bunch of loaded freight cars.
Dig a hole at one corner of the site to get this surcharge, move it over the site in a "U" pattern, bringing the earth back to its source after traversing the site. You would not have to wait at any location, since this is something like compacting earth with a roller.