The Hantek 6022BE measures analog waveforms, it is just always set to be DC coupled and can't be set to an AC input coupled mode. The lack of AC coupling would be a severe problem if you ever wanted to look at noise/ripple on an SMPS.
Depending on the output voltage of the inverter, one thing you might also consider will be the maximum voltage ratings of the inputs: the 6022BE has a max input voltage of 35Vpeak (350V with a 10x divider probe) and the DSO203 is rated to 40Vpeak (400V with a 10x divider probe).
Another thing to consider if you ever look at digital signals will be the sampling. The 6022BE has a sample rate of 48Msamples/s with 20MHz analog bandwidth and a 1M sample data buffer. The DSO203 has a sample rate of 72MSamples/s with 8MHz analog bandwidth and a 4k sample buffer.
Similar to the DSO203, I have a "D602" pocket oscilloscope that uses MCX connectors on the probes instead of the usual BNCs, and one huge design flaw with it is that the probes have no compensation trimmer for the 10x mode. This makes waveforms look heavily distorted, making it virtually unusable in 10x mode. I don't know if the probes that come with the DSO203 will have this problem or not.
I also have a Hantek 6022BE, and while it works well for certain things, the software interface makes controlling it very cumbersome, and the software triggering is very unreliable. The 6022BE and the D602 are nice if I want to bring a scope with me somewhere, but neither of them would be my first choice compared to a bench scope.
Are the Hantek 6022BE and DSO203 the only options you are considering? If you plan to keep using the scope after this project, what kind of features do you think you would want to have?