That's a dilemma. A weak signal, especially for mobile data, is notorious for sucking battery. I would suggest turning off mobile data (4G) at work and just try relying on the wifi, but if both are lousy, then you might be stuck.
One possibility would be to force 3G instead of 4G while at work. 3G networks are usually more widespread, have a stronger signal, and eat less battery; however, they are significantly slower. Still, it might be better for your husband at work. I don't have a Note, but from what I can tell on the web, you can force 3G by doing this:
1. Settings/Wireless & Networks--More.../Mobile Networks/Network Mode
2. Choose the setting that says CDMA (or WCDMA) and that does NOT say LTE. (CDMA is 3G, LTE is 4G on Verizon.) If you don't use Verizon, then it'll probably say GSM instead of CDMA.
There are apps that provide a toggle widget to toggle between 3G and 4G, but they don't always work for every phone. I've seen it mentioned that
Extended Controls has a toggle that works well, but I haven't tried it myself.