That probably depends on the details of how the power in, charging circuit & battery are functionally integrated. If the battery & charge circuit of switched to the side when the power is on, and the battery is switched to (as is done with UPS) then this side channel may still work.
But your comment does point to the fact that this side channel attack only works on modern PC's and equipment because of the use of under filtered switching power supplies.
If you used an "old fashion" power supply that used a transformer, full or half wave bridge, and (optional) analog voltage regulator, you have to follow that bridge with a large (several hundred microfarad to several thousand microfarad) capacitor to put the 3db point for line ripple suppression well below the mains frequency (50 or 60 Hz).
That 3db point "works both ways". Most of the signatures these guys show have most of their info out to 300Hz, and above the mains frequency. And old fashion power supply, with it's 3db point at say, a few Hz, would leave them with basically no noise to analyze. The down side is that old fashion power supplies are bulkier, more costly & less efficient than switching power supplies, which is why they have been replaced in the last couple of decades.
But if you wanted your equipment to block this side channel, just design using a properly designed old fashion transformer based power supply with whopping capacitors.