The soil mix I use for container gardening is Al's mix; it's 2-3 cubic feet pine bark fines, 5 gallons each of vermiculite and spaghnum peat, 2 c of Osmocote (pelleted fertilizer for container gardening) and 1 c lime (dolomite, not fast-acting).

Above measurements yields about 30-35 gallons per batch.
beans
The mix when I mixed everything together was very coarse and looked like regular mulch; the pieces of mulch are too large.

Not everything hates the coarse soil mix, with the pine bark mulch used straight out of the bag, which is which is how I use the pine bark mulch to make it. Lettuce, cabbage, beans, pak choi, lemon balm and oregano love it this coarse. Very large and healthy plants when harvested.

Any root veggie, any squash (winter/summer), melons, tomatoes, peppers and eggplant hate it because it dries out too fast, doesn't stay consistently moist. Root veggies don't bulb/form roots due to being it being too coarse.

The main questions I have are....

The problem is the pieces I have are too lg. I was told I have to screen the mulch which I didn't know I was supposed to do; how do I do this? I don't know hot to, as I've never done it before. It's recommended the pieces be 1/4" to 1/8" in diameter. Can I just shred the mulch, or would this make it too small?

I'm looking for a balanced fertilizer that is higher in phosphorus and potassium for root veggies/fruiting crops (toms, peppers, squash etc.).

Unfortunately the ones that are more balanced aren't suitable for container gardening or it's just granular fertilizer you mix with water, and apply the liquid at the surface of the soil. The only other thing available is a dry fertilizer, but it's something you'd spread out in a plot in the ground over a lg. area and it's also not suitable for container gardening.

My last question is I realize this mix is designed for container gardening, but could you use it in raised beds? I have some raised beds I bought that I plan to use next season, that's why I'm asking.

here is a link to a short thread showing screens (these are Al's) used to screen components for the mixes.
You don't have to make them same if you don't have time/means, or materials, or are in hurry. Anything with appropriate hole size could be used: plastic or metal storage trays, kitchen sieves/drainers, plastic nursery trays and more.

see them here

If your bark is too large & you have shredder, run it thru it. There will be some smal pieces, but you can use them too, if too small for mix, you can mix them into your raised beds. Other suggestions were to run over the bag of bark with your car, or chop it with lawnmover (work over hard surface like a driveway & put a bag over the mulcher).
As for other of your questions, I hope someone more qualified/experienced will answer. I read thru Al's postings about fertilizing & follow that advice - and I am getting great results.

about fertilizering containerized plants

Some of the things you can use to screen:

Where can I talk to Al if I have questions I want to ask him? Couldn't find anything on container gardening forums/threads, and yes I've looked regarding the soils, fertilizers etc.

Regarding the screening, what else would I use besides the screener themselves? Would I place the screener on top of a container? Should I place something below the screener to catch the larger pieces?

Regarding the size of the pieces, what is considered too small? Mix says to use 1/4" to 1/8" pieces but would 1/16" (or smaller) work or would they be too small?

The fertilizers recommended are all liquid.

Can you add the liquid fertilizer to the soil mix, when you make it up? I always thought that doing that wasn't a good idea, and liquid ones were designed to be use at the surface of the soil? Could use clarification with this.

click here

these would work very well.

I already suggested few things you can use to screen (photo in previous post). Also kitchen sieve would work, metal or plastic mesh baskets/trays - use your imagination.
You should screen over some kind of container or a plastic sheet or even wheelbarrow. Anything that would pass thru (fall thru) an insect screen (like one used on windows), is too small to use in container-put it on your garden beds.

I think you should take time to read what's already posted here about soils & fertilizing. Did you have time to read thru thread I suggested? There is so much info available, just have to find time to read.
These are few that helped me tremendously:

read if interested

read me

read me too

...and many more.

There is so much more to read, all it requires is your time/interest.
Al is a longtime member of this forum, he posted numerous posts & answered probably thousands of questions.
There are other members with great answers here.

Liquid fertilizer should be mixed with water for regular watering, don't mix it with soil. It's also explained very well (much better than I can) in threads I gave you link to.
Rina