It is not that you are "splitting" the branch circuit, it is what loads are permitted on the circuits. As you have a dryer outlet, intended for a 5kw dryer, which will draw approximately 21 amperes at 240 volts. The dryer has a motor so that changed the way you calculate the load per NEC 422.10(G) which will send you to article 430 for motors. That article, per 430.53 now must take the actual motor in the dryer at 125% plus 100% of the other loads which will bump that 21 ampere load up a little. Can't tell you how much because I would have to see how large the motor actually is. None the less, we are now looking at a load of at least 22 amperes now.
Normally you would size per NEC 210 but, 210.2 basically states you must comply with table 210.2 which lists other articles which amend or supplement the article. So now article 430, per article 422 drives where we are going. So, you technically would be allowed to add another outlet on the circuit provided it does not draw more than 6.4 amps of continuous load or 8 amps of non-continuous load provided.... the circuit breaker was listed to operate at 100 % of its capacity, which it will not be per NEC 210.19(A) the exception so, we are back to a 30 amp circuit, @ 80% of its rating, being allowed to feed a maximum of a 24 ampere load. Now we are limited to adding no more than 2 amps of load on the other outlet. Bottom line is, it is not worth it but, technically legal.
If you wish to incorporate a transfer switch into the mix, that would be possible but expensive.
Now, all of this said, is it possible? Yes, if you don't comply with the codes, don't get a permit and disclose that you did this modification to the building without obtaining permits on the disclosure form when you sell the house. All of this said, if YOU KNOW what you are doing, perform the work in a mechanically sound manner and do not supply more than one piece of equipment at a time then yes, you can make it work safely.
Lots of ifs in this and again, not to keep harping on this, there are significant hazards due to possible shock and fire. The last thing to bring up is your homeowner's insurance. If you perform this work without permits and inspections, and it causes a fire, the insurance company does not have to cover the claim. Yes, I have seen this and it is UGLY! Especially if you are not using one of the big three insurance companies (State Farm, Allstate and Farmers). People have lost everything, including family members. I have seen way too much of this and it is not worth making it work, the risks are too great.
Sorry for the long post!