It is difficult to say since we can't see what is on the bag.
Does it mention that it is good for raised beds?

A blend of a good quality potting mix and compost (50-50)
is good. Too much peat moss will make the mix acidic.
If you aren't sure you can always add a little more compost. More than one type of compost is recommended. You have mushroom compost already.

It does sound like a good mix if the proportions are correct.

No it is nothing like Mel's Mix. Mel's Mix is 1/3 vermiculite or perilite, 1/3 peat moss, 1/3 compost that is made from 5 different types of compost. The mix your husband purchased has some peat moss in it but we don't know how much. It only has one type of compost. If you want it to be more like Mel's Mix, add vermiculite/perilite and some different types of compost. I'm not saying your soil will not work but you asked if it was like Mel's Mix and it is not. The fir bark may work similar to vermiculite but I don't really know.

I would add coarse vermiculite (I had to go to a nursery to find this... big box stores don't carry it or it is in TINY bags) and maybe some extra compost varieties. You could then do a pH test (kits available at any hardware store or nursery) and if it is alkali, you could ass sphagnum peat moss. That would get you closer than the current mix.... and thanks for making me feel better to micromanage my hubby and buy the soil myself, lol!! I figured he would do something like this, too, hahaha! :)

I second the vermiculite additions as it will only help with water retention. I would stay away from perlite as Mel and many reviewers on this site will tell you it is worthless. I did my first sfg with a potting mix just like you are describing and it did fine. This was before I had even heard of sfg and I had not read the book so I didn't know any difference. Anyway, the potting mix you are mentioning sounds like a pretty good mix. In fact from what i understand, Mel's mix once incorporated the use of sand a good drainage additive. I suspect that the sand was added to this mix to prevent compaction and all that. The "forest material" is a standard additive to most organic potting mixes and helps keep it aerated and adds organic matter as it breaks down. It sounds like the mix is already balanced. Adding a bag of black cow would probably help with extra nutrients. Does it say "feeds for 3 months" or something like that on it? if so, it probably already has time release fert in it, so you will not necessarily need to amend with compost.

Potting mixes are usually bark, peat, and perlite. Add some compost, and it'll be fine, as potting mixes usually drain quite a bit faster than garden mixes.

I wouldn't add vermiculite, as my experience is that it turns into mush once it gets wet a bunch and isn't a useful addition.

That's funny what you are saying about vermiculite JoppaRich. I never had an issue with it turning to mush, although it does act like a sponge. Just make sure you are using coarse ground.