I have had the same issue with my Polar H7 Bluetooth HRM Chest Strap. I've figured out that the Watch will connect to the chest strap if Bluetooth is switched off in settings on the iPhone. Of course I don't want to have to turn off bluetooth on the phone during a workout. I have been using my Polar H7 for about a year now with RunKeeper on my iPhone (and I've also used it with the Polar Beats app and all the treadmills, spinbikes, ellipticals and even the new Jacob's Ladder cardio machines in my gym) and it has worked seamlessly. I can even connect/record HR in multiple apps at the same time on my phone and still see my heart rate (from the chest strap, not the hand grips) on the cardio machines.
I had noted that the wrist based HRM on the Watch was not quick nor accurate enough for my interval runs or spinbike sessions where I am running or spinning for 2 minutes at 70-80% (130-140bpm) and then sprinting/grinding up to 90-100% (160+ bpm) for a short 30 seconds to a minute. So I wanted to connect my H7 to the Watch to take advantage of its accuracy. The first time I connected the H7 to the Watch, I was able to do so easily and had a great workout in the gym. It found the bluetooth chest strap and paired and connected without a problem. After that and for the next week, I was unable to get it to connect to the Watch although it still showed up in the list of Health Devices on the watch, just as "not connected" even though it was connected to my phone and it also connected to the cardio machines, just not the Watch. It was like the Watch couldn't see the H7 even though it was obviously working because all the other devices were still seeing it and reading the HR. Like you, I tried putting the Watch on the charger (wasn't sure if it did some kind of restart when charging) forgetting the H7 in the Watch's bluetooth settings and re-pairing it and I tried restarting the phone and eventually doing a reset on the Watch, all to no avail.
Finally today, I happened to try turning off Bluetooth in the settings on my phone and the Watch immediately connected to the H7. Thinking back, I believe I had also turned off Bluetooth on the phone that first day that I used the H7 with the Watch because I was trying to see if the Watch would use the gym's Wifi. I had just forgotten about that part of the equation.
I called Apple phone support to see if there was a work around, but the support person I got was not "up" on the whole idea of connecting Bluetooth HRM's to the Watch. At first she tried to make it sound like the Polar people needed to come up with an updated version of their device to work with Apple's new technology, but I told her that since that particular device seems to be the most universal one out there as far as working with most phones, apps, cardio machines, etc. and because the Apple Watch is obviously designed to be able to connect to such devices, given that it can connect to it, only under certain circumstances, it would seem to me that Apple is the one who needs to come up with an update. I understand that this is version 1.0 of the Watch and there are going to be bugs to work out. That's a given. I just wanted to make sure that the development team was aware of the issue. Hopefully they'll address it in one of the earliest updates.