Yes, that could definitely account for less battery life. It could also account for better battery life. It depends on a lot of factors. Can get complicated.
I have a 3G Network Extender. I also have somewhat weak overall VZW native coverage in my house, as you probably do, (or none!) which accounts for the reason I purchased a Network Extender, of course.
Anyway, I can stand near my NE and watch my phone switch to 3G (CDMA RevA - verified by Menu | Settings | About | Network). This is because the Network Extender only supports CDMA RevA 3G - not LTE/CDMA 4G nor CDMA eHRPD 3G.
I can get away from my Network Extender and often can pickup a local Native cell tower and get either CDMA eHPRD 3G or LTE/CDMA 4G - all Native signals.
I can watch as my phone switches between 4G/3G on the Native sites as well as when it switches to the Network Extender. Very interesting to walk around my house monitoring this. It also shows pretty plainly how well well the ThunderBolt switches between signal types. This is the same thing that can happen as you travel along outside in a vehicle.
It's kind of like having your own personal little "test lab" in your house. Very interesting.
I have been able to verify on my device using these methods that with MR1 and MR2 there were definite improvements in the "data dead time" during the phone's switchover from one signal to another. This has been the primary Achilles heal of all 4G capable phones - the dead data time between switching. Keep in mind that even when you're not moving the phone is going to "check" on a periodic basis (I don't know what it is) if there is a better/stronger signal present that it should be using. So switches can occur even when you are standing still. Or, at least this "effort" to "check" will continue to go on no matter if you have an existing 4G signal present or not.
Having said all that, I think you can imagine how having this Network Extender in close proximity (but with very limited range, measured in feet rather than miles like Native towers) and you also have Native tower signals present (of any type, 1x/3G/4G, etc.) it tremendously increases your odds of the phone spending much more time "hunting and switching". That equals much more battery use.
Obviously, if you have a very poor Native signal and if you interested in maximizing the phone's performance, you should use Wifi at home for data. This will eliminate all that "hunting/searching" that is so detrimental to the battery. Also, if you always have your phone within a few feet of your Network Extender it should pickup 3G well and maximize battery life on 3G. However, I think you said you only have a 1x Network Extender (older model - for those not familiar). So with a choice of 1x verses a Native signal, it might be working very hard to do better than 1x - unless you park yourself inches from it. I dunno...
Personally, since when I am at home I have constant access to a charger, I don't worry about the battery because I am more interested in being able to gather knowledge about my local Native tower coverage/strength, etc. So I usually leave my ThunderBolt on RF. Just for academic reasons.
-Frank