Hmm... well, I don't know the particulars about the straw bedding you've got there but I can tell you the experience here on our farm and what I know about composting horse manure, having done it for the past 10 years or so. We also compost rabbit, chicken and goat manures here.
Horse manure definitely needs to be composted. A horse's digestive system is pretty simple and weed seeds can survive it, though I honestly don't see a lot of weeds germinating in our horse and donkey manure.
The manure itself has a good carbon to nitrogen ratio and composts well on its own. Left to its own, it breaks down really well without much other than keeping it moist.
Bedding will generally increase the carbon part of the equation and require additional nitrogen source. We sometimes add grasses, chicken manure (which is really "hot", meaning it's got a high nitrogen content and definitely needs to be composted) and kitchen scraps to it to boost the nitrogen. Anytime I've got manures mixed with bedding - wood shavings, sawdust or straw, I always ensure that I add a lot of nitrogen to help things move along more quickly.
You can do the lazy composting method - no turning, relying on more anaerobic composting and it'll take longer for both the manure by itself and the manure/bedding to break down or you can engage in what I call my "farm workout" by turning the pile to introduce oxygen into the pile and it'll be aerobic and typically break down faster - that's been my experience. Keeping it moist and turning it often - weekly or a couple times a month might be fine - will speed things up in both cases.
Moisture helps to encourage the decomposition process. If it dries out, it takes longer, particularly with the bedding.
I don't think my local source of straw uses fungicides and it breaks down reasonably fast.
I try to get my compost bins and piles to contain at least a cubic yard of material as that tends to encourage things to heat up more quickly but it isn't always possible. I'd shoot for something in that quantity (or more!) and see how that bedding breaks down. You might be surprised to see that it does so quickly.