I don't know about transplanting- only ripping out- but the trees will eventually grow above the shrubs and shouldn't be a problem. However, Sweet Gums produce branches over the entire trunk and will need to be limbed up (of course) but over time you will need to rub off emerging branches on the lower trunk.
I don't think the closeness of the plants is an issue.

This is just my personal opinion, but I have grown sweetgum trees in our yard and wish I didn't. We plan to cut them down this year.

re bumblebeez post-
You don't think the sweetgum will crowd out the neighboring plants over time? I know camelias, etc can grow near an oak but suffer if grown near a maple w its shallow roots.

Thanks to bumblebeez and plantnutz for your responses.

I have plants growing around the base of my sweet gums and they are fine. I think the camellias and azaleas won't suffer at all until you decide you can't stand the sweet gums anymore and cut them down!

Sweet gums are considered trash trees for a reason. YOu'll never be able to walk barefoot anywhere in your yard again. Kill them now while it's still easy.

Thanks, hollysprings. I have an acre under hardwoods in largely natural mulch with camellias and azaleas, so I thought maybe I could tolerate sweetgums--but the combined responses/advice of this group is certainly giving me second thoughts !! Maybe I should encourage a few replacement oaks instead.
Tom

We have sweetgums which come up everywhere! They are a real pest. And the frustrating thing is that they don't have to be very big till they are REAL hard to pull up by hand. GRRR!

My sweet gum tree gets covered in asps (stinging catapillars). I would not plant grow this tree because the asps love them.

When we bought this property 17 years ago, there was (still is) one Sweet Gum tree, nestled among an Ash tree and several Pines...I planted Rhodi's and Azaleas, Impatiens, etcs. near and around the trees...and they're doing great.
Though some people complain the seed pods on the ground are messy... (while some people spray paint the pods and use them for decorations)...I think of Sweet Gum as an "innocent" tree...ha...Maybe partly because it brings back some good childhood memories...I wrote in another forum about that...you can read it in (of all places)...
"Garden and Flower Crafts"...Intitled:
An unusual use for Sweet Gum trees"
I like it so much, that when my husband had men cut down a lot of our Pines because they were too close to the house and wires...they were SUPPOSE to cut down the Pines only...but the first day here, they also cut a few small Oaks...(and left us with a huge mess that they had promised to clean up)
Knowing that they were coming back the next day...to cut down a few Pines on the side...and there would be noone at home...I bought two large white posters...and with a black marker wrote on each: "Do Not Cut This Tree Down" and wrapped one around the Ash and the other around the Sweet Gum tree.
Not only were my trees still standing that evening when I returned home, but according to our next door neighbor, AKA Mrs. Nosey...ha... who phoned me...said that the men got mad because I wouldn't let them cut those trees and just left without cutting anymore down, Good!...that I should have let them cut all the trees down...
The trees were free, they probably made a lot of money from the wood.

Sweet gums have a very deep tap root (which is why they are so hard to pull up) and as a matter of personal taste I also vote for killing this tree unless you absolutely need quick shade.

After studying the huge sweetgum that spawns all my volunteers, and seeing my wife's curly-haired dog running around with sweetgum balls on his butt, I cut down my largest volunteers and pulled up the rest.
I have found a SEEDLESS sweetgum variety at a Greenville, SC nursey and think I'll try one. Does anyone have experience with a seedless sweetgum? The nursery guy said it would have purple/red fall leaves which is not as good as yellow, in my book.

Help! We have two huge sweet gum trees that overhang our property. So far everything we have planted within a 50ft area of them has died! Is there something about sweet gum trees that kill other plants??