I have been trying to understand what typically happens to US Treasury bonds during a period like this. A rational train of thought would be that the interest rates on them should continue to go up as the Fed continues to hike rates.

However, assuming a fairly significant stock market crash is on the horizon, doesn't that mean that many investors will look for a "safe haven" and put pressure on demand on them causing the price to go up (and also making the effective interest rate lower)? Which force wins?

Bond pricing and fluctuation prior to maturity are one reason I've never bought a bond. It's also why you need to look at both CY (Current Yield) and YTM (Yield to Maturity) when buying and ask yourself if you'll ever need to sell early. If you're using an FA ask him what his YTB (Yield to Broker) is, sometimes it's up to $20 a bond, getting him to cut that can significantly improve your numbers. Before being a Treasury I'd look at municipal ETF's or CEF's.