The only reason to get a running shoe for what you are doing would be for the treadmill (if you are running and not walking). For everything else a cross trainer would be preferable. Most people don't realize that running shoes are for "running". They are built to run forward and don't have the side support for most other activities.
Its just about 10 minutes of running time 3 days a week during winter season. Looks like it may not be worth it. The annoying thing is sometimes I play racquetball. it becomes hard to decide what to buy/wear when each of these activities seem to have different preferred shoes :-)