Home bath scales at home generally have a 2 pound margin of error. Thus the reading it gives you is generally 2 pounds heavier or 2 pounds lighter than what you really weigh. And that assumes it is correctly zeroed (displays zero pounds when there is nothing on the scale). If it isn't correctly zeroed, the error is more than 2 pounds.
Lets pretend your wife weighs truely 130 pounds and your daughter truely weighs 13 pounds. Lets say you have a typical home bath scale and weigh your wife. With its typical error, it will display anything from 128 to 132 pounds. Now add your daughter. It will then display anything from 141 to 145 pounds. Now lets subtract the two: you daughter could weigh 141-132 = 9 pounds, or your daughter could weigh 145-128 = 17 pounds, or anything in between. Thus your method of weighing your daughter will give you anything from 9 to 17 pounds! This is a very inaccurate method. (1) home bath scales aren't very accurate, (2) subtracting two similar numbers will magnify any inaccuracies.
Your doctor's scale is much more accurate than any typical home scale. Plus your doctor didn't have to subtract (thus he didn't magnify the inaccuracy). I'd trust your doctor's scale more than anything you have at home. It is possible that his scale wasn't zeroed properly, but I doubt that is the case.
Too many people think your weight gain/loss depends on only two factors: calorie input and exercise. Sadly that is far from the truth. The amount of weight anyone gains is the weight of food you eat and drink then subtract the weight of stuff that you eliminate (bathroom duties, sweat, etc.) That is the basic law of chemistry. You said your daughter ate a lot, but if she eliminated a lot as well, the two effects cancel and she won't gain weight.
I don't have much experience with that thermometer. But it was designed and calibrated with the probe covers on. Any other use will not be doing what it was designed to do, and thus will have errors - thus the discrepancy there should exist. But you could still be using it incorrectly, or you could have gotten a bad device. If you want, call the manufacturer.