The overall antenna length being very close to a perfect 1 wavelength is the linchpin of the issue, I suspect. It is not surprising that it can be easily moved outside of the tuning capability of your tuner, as the impedance peak near resonance can be quite sharp (depending on the Q of the antenna system), allowing the impedance to change by hundreds or even thousands of ohms over a bandwidth of just a few percent of the signal's frequency. It is quite possible to see a swing from 10:1 SWR at the tuner (which is well within the range of most decent tuners) at 14.350MHz to 100:1 at 14.450MHz in a well engineered antenna with good quality feed line.
I would wager that if you were to do an impedance sweep of the antenna system during warm weather, you'd find that the actual impedance maximum (indicating a perfectly resonant 1 wavelength) would be just a bit above 14.350MHz, probably not by more than a few tens of kHz. Water, ice, and snow on the antenna can contribute a small amount of electrical loading, effectively decreasing the resonant frequency, in this case dragging the impedance maximum in to the ham band. This is common with insulated antenna wire and 300 or 450 ohm "window" type balanced feed line.
In addition, the larger dielectric losses contributed by the liquid around the antenna compared to air can alter the resistive component of the complex impedance, and asymmetrical dielectric losses on the two sides of the system, especially on the balanced line, can contribute to significant imbalance and common mode current, causing the feedline to act as part of the antenna, which will further alter impedance and cause unpredictable changes to SWR.
Since one end of the dipole is quite low to the ground, just 1/10 of a wavelength or so, coupling to the ground and nearby objects is probably relatively significant. If the recent cold weather also came with a new snow fall, that may further contribute to a shift in resonant frequency.
The point being, you are likely seeing the antenna's resonant frequency shift from somewhere above 14.350MHz to somewhere very near the 20 meter band due to loading effects of water, ice, and snow. It's also possible that the cold is causing mechanical changes to the system, such as sagging that brings some part of the antenna or feedline close to an object it can couple to. You may also be experiencing an increase in common mode current on the feed line, which will result in inaccurate SWR measurements. The BLT tuner should have good common mode isolation with that isolated balun, so I wouldn't be quite as inclined to suspect the SWR meter here.
Exactly how to remedy the problem depends on quite a few variables, not least of all how much flexibility you have with the antenna location and construction. If the balanced line is "window" type line, sheltering it to prevent any rain or ice from accumulating on it (dust can cause problems too), or moving it up away from the ground to avoid problems from snow accumulating around it may help. A better bet would be to ensure that it is kept away from any object it could couple with by a distance of at least the distance between the two conductors, but ideally 2 or 3 times that. Replacing the window line with true ladder line can minimize problems with dielectric losses when wet or covered in ice, and would probably be the optimal solution.
Getting the low end of the antenna another couple of meters above the ground may also significantly improve the situation. As it is currently installed, there is almost certainly some degree of imbalance on the balanced line due to the asymmetrical interaction with the ground between the two ends of the dipole, which will make the balanced line all the more likely to interact negatively with objects around it.
Another option would be to slightly lengthen or shorten the antenna, moving the 1 wavelength impedance maximum away from the 20 meter band. A total length of around 25 meters is often a good compromise, as it will retain good characteristics on 40m, while still allowing it to tune both 20 meters and 30 meters with a reasonably good tuner.