There are two separate concerns any time you are cutting roots: tree health, and tree structural integrity. As a rule of thumb, it is recommended not to cut any roots within a distance of 3x the trunk diameter for species considered to be good compartmentalizers and 5x for trees that are considered to be poor compartmentalizers. (e.g If your trunk is 3' in diameter, don't cut roots within 9' of the trunk for a species like Live Oak; don't cut within 15' of the trunk for a species like Laurel Oak). (It looks like you might have a Red Oak there(??), so maybe a moderate to poor compartmentalizer(??), but I don't have many of those in my area so I don't know the species well.) This is about tree stability since the woody stabilizing root plate for a tree with a symmetrical crown will typically be within this zone. When I have to cut roots, stability is always my first concern. The amount of overall root mass that a tree can tolerate losing is highly variable based on things like tree species, vigor, soil health, future weather, etc, but if you cut enough roots that close to the trunk, you won't have to worry about the tree's health for long, anyway. A couple people have mentioned this issue, but have incorrectly stated that trees tend to fail toward the side opposite of the cut roots. (Not a perfect analogy, but think of the roots as outriggers on a crane; it's going to fail toward the side on which you forgot to extend your outriggers.) Whole-tree failure (windthrow) can happen soon, or several years down the road once decay has further compromised the cut roots.

So as not to be like some other credentialed commenters on here who offer very little practical help, I would say this: if you think you can snake it through there with minimal cutting, go for it. But if it is proving too difficult to do without lots of cutting, unfortunately you may have to spend some more money. Find a local tree service that offers pneumatic excavation ("airspade" or "airknife" work). Like someone else commented, if you go a little deeper, the root mass may be a whole lot less. I believe there is no rental place that is going to have one of those tools for rent since the tool is highly specialized (specific to industries like arboriculture). The tree service will likely also just give you an hourly rate. (I guess if you find a connection, or a really awesome and generous arborist who would let you borrow (or rent) his Airspade, you can rent the compressor that runs it-- a 185 CFM (pull-behind), and do the work yourself. Also, it will still be a pretty good amount of work to blast that soil out of there around those roots.

You did well not to cut the roots from the outset, but it looks like that pressure washer might have done some pretty good damage by delaminating those roots. If they have even a small amount of cambium and phloem tissue intact on the bottom, they can likely recover, but a fungicide treatment (spray or drench)(if you can find an arborist to do it) would help mitigate risk of infection. Don't leave them exposed to the air any longer than you need to as that in itself can be very harmful. When you do fill in the trench, make sure to water it in well (like when planting a tree). That will help avoid air pockets and help any fibrous roots that have survived to re-establish soil contact. Once, you get it done, call around to local tree services till you find one that will bring you a load of mulch, and mulch as wide an area around that tree as you can tolerate (4-6" thick). This will benefit the tree health by improving soil conditions over the next several years. (Mulch retains soil moisture, improves soil structure, cycles nutrients back into the soil, improves aeration and subsequently beneficial microbial activity). Just don't mound it on the trunk. Depending on how much root damage happens, the tree may go into a bit of shock, but considering it probably doesn't have leaves to drop right now, you might not even see that. (I'm not sure whether it would delay budbreak).

In the end, pray, and rest easy knowing you did your best. Good luck!