If you are not after portability but want to avoid any switching noise, why not? It's a 7 Amp*hour battery, not 7 A, and when discharged at 4 A it will give you less than 7 Ah capacity. But should certainly last you more than an hour.
I am not 100% convinced by your cause-of-noise analysis. Can you be sure it's wire-bound immission rather than radiated? Also, if the immissions are indeed coming wire-bound from your monitor -- why not switch that off while you are taking the measurements?
I believe we had someone comparing noise between a mains-powered DHO800 and one powered from a battery, and there was no significant difference. The power supply of the DHO800 itself is pretty decent. (Excluding the 12V fixed supply shipped with the pilot models last summer.)
EDIT: Assuming that you observed the noise while using probes, have you tried using the ground springs instead of the short ground leads?