I cobbled together a proof-of-concept using the 6-30V timer relay module.
The white section of track is the insulated-rail trigger zone. The timer module is powered by 12V DC coming from the left. I used a pair of 14V bulbs to simulate your 3-wire signal head(s). The timer module is set to mode P1.2 as discussed earlier. In this case I used a delay time of 4.0 second though it can be set to anything you want.
Note that I modified the diagram using this timer module to use a diode; you mentioned you have diodes in your "stash." One diode can handle two (or more) modules.
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Separately. Yes, the math works for your 22,000uF cap method. That's 22x the energy storage of a 1000uF cap so should yield a delay of several seconds which is apparently what you get! Amazing. 
I see I did not identify the inexpensive 6-30V relay timer module example as being from Aliexpress.com. 2 modules for ~$6 shipped...but means a "new" vendor so unclear if worth the bother to save a couple bucks if you're already set up on eBay. I've recently started using Aliexpress (Asia) and have had no problems getting electronic components and modules so far with shipping time about the same (2-3 weeks) as from eBay-Asia sellers.
