You won't have much accuracy at all doing it that way.
I would recommend you use a
MAX31865 instead, with its voltage or ground switched via a transistor or MOSFET on demand by Teensy.
You can find cheap (4-5€) break-out boards on eBay for example. They typically have a 430 Ohm reference resistor, but for PT1000, a precise 4000 Ohm resistor would be much better. (Even just replacing the resistor with one whose resistance you have measured accurately -- it does not have to be 4000 ohms exactly, 3.92k or 4.02k will be just perfect; all that matters is that you know or measure its accurate resistance, so you can do the math on Teensy correctly, giving you accurate temperature measurements.)
Plus, you can use a three-wire PT1000 sensor, and with just a little math on the Teensy (as the PTC resistance as a function of temperature is not exactly linear) you can get half a degree Celsius/Kelvin accuracy; with a fraction of a degree resolution. (Meaning you can detect relative changes, on the order of a fraction of a degree, but with proper math compensation, your measurement can be accurate to within a degree or less over the entire range.)
The MAX31865 does not consume much current at all, no more than 10mA or so during conversion -- that 10mA being just the figure you need to use for its power budget, i.e. using a small transistor with a maximum current rating of say 50 mA is plenty. You do need to power the MAX31865 for something like 10 - 20 ms from power on before initiating a measurement, each measurement taking somewhere around 50-70 ms, so budgeting about 100 ms per measurement should work fine.
I have a few MAX31865's as well as 3-wire PT100 (but no PT1000) temperature sensors, so if there is something you wish me to check, let me know.
I do also dabble with EasyEDA (but I am only a hobbyist bumblespork!), so if you want to make your own board, I may be able to help. The MAX31865 circuit is simple, just three capacitors and a precise reference resistor -- the value itself isn't
that important, as long as you know it precisely! --, so adding a power-controlling transistor with a current-limiting resistor, and maybe an inrush current limiting resistor for the MAX31865 power too, would be a simple thing. I like EasyEDA, because it is easy enough for a hobbyist like me, and getting the boards manufactured at JLCPCB is easy;
here are some of my own "designs". Other members here have much more experience than I, though!