You can do it -- you'll need a subpanel loadcenter in a NEMA 3R enclosure, though, as well as watertight conduit/raceway and fittings for the outdoor runs, which will be done using THWN in the conduits.
You will want to put a 30A 2pole for the dryer and a 20A GFCI for the laundry outlets, as well as a spare 20A breaker (if you want a lighting circuit there, or an extra laundry outlet circuit) in your subpanel, which does not need a main breaker of its own, so it can be a four-slot NEMA 3R loadcenter, such as an Eaton BR48L125RP with a GBK10 ground bar in it, as the neutral and ground will be on separate, isolated bars in the subpanel.
As to the feeder breaker, the NEC calls for 1500VA for the laundry outlet circuit (220.52(B)) and a minimum of 5000VA (a 20A dryer) or the wattage of the dryer (whichever is higher) for an electric dryer load (220.54). Assuming your dryer pulls the full 30A (7500VA) and you have 1500VA of laundry circuit load, this yields 9000VA or 37.5A, which requires a 40A feeder breaker in the main panel, and 8AWG THWN for the feeder wires. In reality, your dryer can't sit there and pull 30A continuously without tripping your breaker, so this is a conservative calculation.
You can't just throw in a bigger breaker, and expect it to work the way you want.
First of all, you're going to have to have the utility install conductors sized appropriately for 200 ampere service. Depending on how the building is wired, you'll also have to upgrade the conductors that run from the service drop to the meter (this might be handled by the utility, depending on how the building is wired). Next, you're going to have to upgrade the conductors that run between the meter and the disconnect (likely handled by installing the new panel pictured). Finally, you may have to upgrade some of your grounding and bonding conductors, as they are typically sized based on the size of the service.
Once all that's done, then you can likely proceed as you've described. However, you're going to have to evaluate how the system is grounded. The service neutral will have to be bonded in the new panel, where the 200 amp disconnect lives. Then you'll have to feed the existing panel (now a secondary panel), with a 4 wire feeder (hot, hot, neutral, ground). You'll also have to isolate the grounded (neutral) bar in the existing panel, if it's not already.
Long story short, this is not a simple job. There are a lot of subtle details, that can easily be missed by a DIYer. I would recommend having an Electrician at least help you plan the project, and inspect the completed work.
Best Answer
First of all, congrats, you did a great job researching before asking the question. That's expected of participants here and you did that.
To answer your question, yes, there shouldn't be any reason you can't add a breaker to the outdoor combi box. Others here might chime in with local code issues, but I doubt it. If it makes you more comfortable, turn off the main breaker to shut off power to the buss. Remove the cover, clip the new breaker in place, twist out the cover for that space and wire up the outdoor circuit using conduit and THHN/THWN or UF (UF is hard to work with!), NM (Romex) isn't legal in outdoor installations, even if in conduit.
I don't want to be a dick here, but this is a pretty basic question indicating a basic unfamiliarity with simple wiring. While this is a DIY Home Improvement site, remember that improperly installed electrical circuits can be dangerous. You might want to grab a book at HD or Lowes on basic wiring principles. Again, I really don't want to be rude or condescending, you might actually enjoy reading about basic wiring and how to do it right. Take care and stay safe.