Many years ago I got an electronic door bell designed to have an extra speaker connected and two door pushes with a different tune played for the back door. So the wiring was supply from transformer, connection to front door push, connection to rear door push, and connection to second speaker 8 wires in all. Over the years the white plastic went yellow, I tried to find a reasonably priced replacement, however either they were more than I wanted to pay or were wireless, so I bought three wireless door bells from Lidi and programmed the same door push into all three. One placed front of house, one in kitchen, and one upstairs. The door push button cells last a long time, did get some water in the push, but clearly two spares now on second. And because three sounders even when one pair of AA cells goes flat we will normally hear the other two and change the batteries on the one with discharged cells.
I do think the originally hard wired was better, however my wife is into craft so spends a lot of time upstairs, so the third sounder upstairs is good.
My late mother the problem was in a wheel chair it took too long to get to the door, here we went for a different method, the large door bell can be either battery or main powered the latter needs a 12 DC power supply, this is what we used, the door push is rather large, it could hold two D cells, the sounder looks like a cordless phone, and pressing a button on the sounder allows you to talk to who ever is at the door, it even has the option to expand it, and be able to release the door from the sounder. The instructions say you can in fact use a cordless phone as the sounder, I did try but it would not work with mothers BT 6500 phones. Like any cordless phone the sounder sits in a cradle so batteries never get discharged. And like any cordless phone you can carry it around the house with you.
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One does have to be careful with wired door bells, the electronic type use very little power, and having two connected together is likely not a problem, but the ding dong door bell often has a huge load, in fact more than the transformer is rated for, it relies on people not pressing the button for long, fitting two of this type you will likely have a problem getting a transformer big enough, volt drop on the wires, and door pushes burning out.
I think you will find you have no option but go wireless, however that does not mean it has to be battery powered.
