Would more shaving help with that or should I just put a radiant heater in there?
Where are you located? Most folks on here are in agreement that chickens don't need supplemental heat in the winter. Of course some breeds are more winter hardy than others. My girls have survived temps down to 10 degrees without heat, but I do have some smaller, large combed breeds (leghorns) that I worried about frost bite, and added a heat lamp a couple times when nighttime temps were -5, which is very rare.
My neighbor is old school and uses a rickety old uninsulated
and unventilated coop with about 40 chickens and she has never used heat, nor has she lost any to the cold that I am aware of.
Chickens molt (lose their feathers and grow new ones) to prepare for winter. After molting, their body takes a break from laying.
Light triggers them to start laying again. When there are 12-14 hours of daylight, a hen will usually resume laying. Some breeds may start a little sooner than others.
A pullet hatched in the spring may start laying for the first time in the fall. I've had them start as late as November laying when they hatched in May.
A broody hen who reared a fall hatch may also start laying again. Broody hens usually molt after each hatch, although not as dramatically as a fall molt, and so she won't begin a fall molt. A broody will also go without laying for a minimum of about 7-10 weeks from the point of beginning to sit on eggs, to the end of weaning herself from her chicks.
If you want to trigger your hens to lay, light and not heat is how you do it. It is important that you give a
minimum of six hours of darkness. This is the standard by the commercial poultry industry for ethically sourced eggs. It is also important that the supplemental light be given in the morning, and not the evening. If they try to go to bed at night and you flip on a light, in affects their sleep, and can also created issues with returning to the coop at night.
If you provide light and your hens start laying, you will shorten their lifespan. Statistically speaking, hens who lay more, die sooner. There are reports of hens (Google Matilda, the world's oldest chicken) who have never laid but lived to be 16.