Tony, I haven't any experience with sidecars so I'll just put my thoughts up until a few of the outfit boys can give you some proper advice.
Costwise, it'd be a huge advantage to fit a car tyre as they're much cheaper and last a lot longer than a bike tyre. Mind you, a lot of your decision will be affected by whether the tyre leans over as you steer, as this will chew out the outer edges of the tyre before the centre.
As for your heavy steering issue, is that while motoring along or when you're trying to maneouvre into a parking spot?
Sometimes, just putting more air pressure into the tyre will make for lighter steering, though you can only go so far with that option.
I stole a closeup pic of your front end for a comparo with this next pic:
^ This is a shot of the leading link on Steptoe's bike when it was a tripod. Notice that there is an optional position for the link pivot point to adjust the amount of trail which directly effects the amount of leverage over the front wheel. Maybe you could get your link modified to offer a couple of options there?
I also have to point out that you have quite narrow bars which will also contribute to the heavy steering.
Other things to consider are that your current front tyre looks like it would offer a lot of grip, whereas a car tyre in the same situation would probably give less grip despite having a wider footprint on the road due to the generally much harder rubber compounds used in them, which would actually help to lighten the steering somewhat.
