Calibration with or without temporal structure Setup
I have a sensor which measures some air property outside. It outputs values on a continuous range. The sensor is expensive, so I want to replace it with multiple cheap sensors, which may give less precise measurements. My goal is to create a statistical model to calibrate the cheap sensors to the output of the expensive sensor, which I define to be the 'correct' value.
Map interpretation
Based on this setup, I can view my goal as finding a map $f$ between the, say two, predictors $x_1$ and $x_2$ and the output $y$.
$$y \sim f(x_1,x_2)$$
Viewed in this way, I can construct scatterplots which indicate that, indeed, the values of $x_1$ and $x_2$ contain information about $y$.

If, at some point in the future, I obtain a new value for both $x_1$ and $x_2$, I can use my map $f$ to predict the value for $y$. A straightforward value-to-value calibration.
Time series interpretation
The original measurements where taken over the course of some time period. Let's say one year. It turns out that there is a lot of temporal structure in both the $x$'s and in $y$.
Now my goal can be restated as saying that I want to predict the entire sequence of $y(t)$ based on the entire sequences $x_1(t)$ and $x_2(t)$. Note that I am not trying to forecast, or to extrapolate beyond this fixed time range. I just want to map sequence-to-sequence.

Viewed as a time series, $x_1(t)$ includes seasonal variation. Both at a 24 hour frequency and over the course of the entire year. The time series of $x_2(t)$ does not show seasonality, but does have persistence. This means the values change slowly over time.
Seasonality
For each data point, I know the time of day and the day of year. So I may include seasonality in my model for $y$. For example, I could add some periodic function $s(t)$. As the time of day and the day of year provide additional information, this can help predict $y$. Then my regression becomes $y \sim x1 + x2 + s(t)$.
Persistence
Given that $y$ changes only slowly over time, I can exploit this. For example, I can use ARIMA-like terms for $x_2(t)$. Then my regression becomes: $y \sim x_1(t) + x_2(t-1) + x_2(t) + x_2(t+1)$.
Philosophical Question
My question is a purely philosophical one. My goal is to calibrate the cheap sensors so that they give the best approximation of the value of the expensive sensor. Adding temporal information to my prediction will likely give a better estimate of $y$. However, does this count as calibration? Time or season do not cause air quality, though they may correlate.
Once the cheap sensors are calibrated, I will use them to gather more data. This data should approximate whatever the expensive sensor would have measured. So my aim is to make my statistical model generalisable. But which view leads to the most general model? The map view, discarding temporal information, or the time series view? I realise this is a weird question, but: what should I want?
Further considerations
Even if I exclude explicit temporal information, $x_1(t)$ includes seasonal variation, i.e. $x_1(t)$ and $s(t)$ correlate. So if I want to find $y \sim f(x_1,x_2)$ and not $y \sim f(x_1,x_2,t)$ by excluding $s(t)$ from my regression, wouldn't my regression simply extract the seasonal effect from $x_1(t)$, giving me more-or-less the same prediction? I'm not even sure if this is a good thing or a bad thing?
Code
Although the code which generated the plots should be irrelevant, I include it here for completeness. Note that my question is not about code, nor implementation, nor mathematics. It's about philosophical considerations when hypothesizing a mechanism.
set.seed(123)
timemax <- 24 * 365
timesequence <- seq(1, timemax)
e1 <- rnorm(timemax)
x1 <- 1.0 * sin(2 * pi / 24 / 365 * timesequence) + 0.4 * sin(2 * pi / 365 * timesequence) + 0.2 * e1
smoothing <- 100
e2 <- rnorm(timemax + smoothing)
x2 <- rep(0, timemax)
for (i in seq(smoothing)) {
  x2 <- x2 + e2[i:(timemax + i - 1)] / smoothing
}
e <- rnorm(timemax)
y <- 1.0 * x1 + -4.0 * x2 + 0.1 * e
par(mfrow=c(1, 3))
plot(timesequence, x1)
plot(timesequence, x2)
plot(timesequence, y)
par(mfrow=c(1, 3))
plot(x1, x2)
plot(x1, y)
plot(x2, y)

 A: You are making the hypothesis that a hidden variable drives the measurements of all your instruments. The measurements of every instrument, good or bad, are indirectly correlated to each others.
However in the code you provide, you hypothesize a direct relationship between your best instrument, and the two others. That is conceptually completely different. Although the methods to uncover the mapping from the bad readings of your cheap instruments to the "true" value of the expensive one, are the same...
That being said, as a first step, imagine you have N bad sensors Si (i = 1 to N) that sense temperature T every minute over a year.
You also have one unique and very expensive sensor R.
The true temperature is unknown. However it is that true temperature that drives all your measurements from all sensors Si and R.
R gives the reference temperature Tr.
Si sensor gives the bad-but-still-a-little-bit-informative temperature Ti
Imagine you take seasonality into account in your model. Then your model will use that information to predict Tr given Ti and season. Why not ? Maybe that can be very successful for prediction of this year Tr. But if some major climatic event occurs next year, that renders the seasonality information incorporated in your model useless or worse misleading, then the predicted Tr will be of similar usefulness...
Then, IMHO, you should probably not incorporate season in your model because the only relevant information you will have to predict next year's Temperatures will be your cheap sensors' Ti (and season will be irrelevant because it may be highly perturbed by a major event next year in my story).
A: No you should not include seasonality into your model, because then you are not asking how much variance of the expensive sensor can you explain using the cheap sensor, but you are asking how much variance of the expensive sensor can you explain using the cheap-sensor and seasonality data, which is obviously not what you want. 
If you only care about R2 or some other metric, than you don't have to do anything else, however the problem is that due to seasonality your data are not IID and thus confidence intervals and p-values will be incorrect. In that case you will have to model the seasonality or other dependence or just detrend the data.
Ofcourse oyu might want to do some additional analysis to check if your sensors are nto only similar in the winter 
