Using Machine Learning to Predict Temperature
I would not recommend going ahead with the data that you know might be wrong.

Looking at your current data, the reason you got a bad result with linear regression is because the relation between them is not linear for the current data. For eg. There is high variation in response (i.e. user_set_temp)  for same value of your predictor (i.e. environment_temp).

First, get hold of correct temperature for your recorded timestamps from local weather data to replace environment temp with this data, till you get the issue rectified with your original recorded environment_temp. When you rectify the issue, then I would recommend you to use both weather and environment temp to predict, as a person might set the temperature depending upon a combination of both. 

After you get a good representative data, this should be a reasonable procedure to help you predict the temperature:


Exploratory data analysis for uni-variate timeseries:
This will help you observe patterns in data, which will help you decide on which new features to engineer/ create from timestamps (step 2) you have of recordings. Also, look at the acf/ pacf plots to help figure out the lags in data that might help you predict better. Fit a basic uni-variate model like STL or ARIMA model to get a good base model for prediction (this model you will compare new models with to see how they perform)
Feature Engineering: Time of day, day of week, week of month etc identified in step 1. Also, look for any other features that might help your algorithm find patterns.
Regression Models: Test with advanced algorithms like RandomForest, Gradient Boosting and RNN-LSTM to check how they perform against each other and the base model.