Don’t train on the LFP. It’s just not worth it at speed. Too big of a risk with very little reward. If you need to go on rides there, go south past the museum campus.
For routes, Chicago doesn’t have hills, but it does have wind. Plan your rides into it.
If you can’t leave the city, there are a few other options. You can use estabilshed race courses and inner roads of parks during early morning. For example, the training crit loop at LaBagh woods, Lake front crit at montrose park or using the Humboldt park inner road. Furthest option would be Calumet park.
The other option is using the diagonals. You can get a pretty good pace on Ogden both south and north. Or you can use what was called The Knife when it was messenger race Milwaukee to Elston, Elston to Milwaukee.
Last option, if you are on a cx bike or gravel bike, you can set up a course in Humboldt or Montrose.