Here is my understanding of the issue.
A 10:1 probe is a complicated 1/10 voltage divider.
You want to apply a 350V AC RMS voltage and look at the ripple.
That means you are applying 350*2*sqrt(2) AC peek-to-peek at the probe tip, or 990V -- Let's call it 1KV.
Then, 1/10 of 1KV is 100V -- well bellow the scopes spec of 400V.
p.s., make sure that there is no way the ground clip on the probe can be disconnected while probing high voltages.
No, the OP wants to look at low level ac "hum" riding on top of around 350v DC.
I will, however, address your comment:-
Peak to peak isn't a real thing with ac signals, as the positive & negative half cycles do not exist at the same time.
The highest voltage at any time is the peak voltage on any half cycle---with your values, about 495v.What I believe the OP was worried about was that the coupling capacitor in circuit in the "ac coupled" mode could charge up to the full 350v DC voltage over approx
5t=CR.Here, it is assumed that
"R"is made up of the series resistance of the probe, in turn in series with the input impedance of the 'scope vertical input,(assuming the latter to be a pure resistance.).
This further assumes that the coupling capacitor's DC voltage rating is less than 350v.
Once the cap breaks down, possibly, all sorts of nasty stuff can happen.
As the 'scope is being used to look at low level ac, the volts/div will be set to a quite sensitive range,which will be overloaded by the 350v DC, with probable damage.
Against this, is the fact that, over many decades, people have used analog Oscilloscopes set to "ac coupling" to look at low level ac superimposed upon DC voltages of the same order as those the OP has.
If you have a fairly sophisticated analog 'scope, it will have adjustable "DC offset", which allows you to stay "DC coupled", but most of the analog 'scopes used for this sort of thing don't have that function.
If damage in the way anticipated was common, it would have become part of the "folklore" of the industry, but it hasn't.
Possibly this is more of a problem with DSOs.
I have read a lot of warnings about this as a problem, but I have not heard of a lot of people killing their DSOs in this fashion.