Has anyone shipped clothes?

Yes, I have.
You don't say in what country you need your business clothes. No matter, if you need something to be in place waiting for you at a certain destination, you are risking it getting caught up in customs and you would have only a bill waiting for you rather than your box. I often see complaints about these customs surprises from other expats on the expat forums I belong to.
However, your safest bet would be using Send My Bag because they follow the shipped goods all the way to the destination. They require itemized inventories with estimated values for each item. If your shipment is caught up in customs, you would know in plenty of time. By the way, they use DHL and FedEx. Good luck.

Edit to distinguish shipping to Europe vs from Europe: shipping to Europe is trickier than shipping to the US. It's got to get past customs without you getting dinged. The US is more lenient.

Yes, I overpacked on a trip to Germany last year and wound up shipping a big box of clothes home via DHL. It was quite easy. I bought my packing box at a DHL post store, printed out the customs form and then took it back and they shipped it. It’s not cheap. I think I had a 30 pound box and it cost me around €45.

But it was really nice to get rid of it. It took about three weeks to get back home to me in the states and everything was intact.

You could use your business casual clothes for traveling too. There are shoes that work as either business or casual, and the shirts & slacks can work too if you pick them carefully.

Would you have time in your trip to shop for 2-3 new dress shirts and a pair of nice shoes? And are you a size that’s typically easy to find clothes off the rack? If so, you could bring your backpack with a lightweight packable duffel. Either place the duffel inside the backpack or the backpack inside the duffel. After your last week, load both items up to come home.

I like to shop when I visit my relatives in Iowa because the shopping in my city is very limited for the styles that look good on a retired age person. I do the above to bring everything back home.

It might really make a difference. For example, Mardee said she was able to ship about 30 pounds of clothes for about €45 from Germany.

For €45 from France, you can probably ship about 3 ounces of clothes. Ok, maybe I am exaggerating, but not by much. Shipping between the U.S. and France is prohibitively expensive.

So it really would help if you told us which country you have in mind (or maybe you are thinking more generally, as you often have work trips to different countries ? Are you at least able to list the involved countries if more than one?)

Thank your for all of your ideas and suggestions. As for locations, my events are always in either Frankfurt, Munich or Madrid. Frankfurt is usually the most convenient and cost effective arrival city for me from the US. But any American Airlines served city from DFW, Chicago, or Charlotte works as well.

I hadn't though about the idea of shopping and then taking my new clothes home. Costs would be similar to the shipping or storage charges that I have found and I would still have the clothes! Will definately do this if I splurge on an authentic Bavarian Tracht to wear in Munich.

Well, if you decide to ship clothes home, should definitely be able to find many DHL locations in either Frankfurt or Munich. Here is a site where you can search for locations: https://www.dhl.de/en/privatkunden/dhl-standorte-finden.html