I used to use TM902C handheld digital thermometers (pyrometers) from Ebay. They read DegC only and only work with a typeK thermocouple. One disadvantage is that you need to convert if you think in degF. An advantage is that there is no way of having it set on the wrong range. Another advantage is that it costs under 5 bucks delivered.
I had maybe 20 of them over 4 or 5 years, bought in ones and twos. Every one went on the calibrator at work and they were all as accurate as western-built (US, UK, Germany, etc) units costing upwards of 20 times the price. I bought about a dozen of them in a single order a year or two back and they didn't look quite the same as the previous units when they arrived. This batch all checked out perfectly up to 800 degC (1472 degF), then started to show an error which increased with temperature. I can't remember which way round the error went at the maximum I could compare: I either had a reading of 1290 degC (2354 degF) at a simulated 1372 degC (2500 degF), or a reading of 1372 degC at a simulated 1290 degC. Either way, it was no use for measuring forge temperatures.
Although I would no longer trust one I had not personally checked at temperatures above 800 degC, they are fine at typical tempering temperatures. They come with a Glassfibre insulated bead thermocouple that is good to around 400 degC, 750 degF and is great for checking accuracy on tempering equipment, being thin and flexible enough to close in an oven door for those using domestic ovens. To me, a TM902C is worth having for that thermocouple alone.
I now use primarily DM6801A units, again off ebay. These are switchable degF/DegC, have proved boringly accurate on the calibrator and are about 10 bucks delivered. Again they are type K only. All mine have come with what seems to be a PVC-insulated bead-type thermocouple. PVC insulation is only good to around 105 degC/221 degF, limiting it's usefulness to us. One of these will work with your Omega thermocouple, assuming it is a type K, though it's worth checking the connector. The link shows a standard (round-pin) connector and you would need to adapt to a miniature (flat-pin) connector to plug into the instrument.
My preferred thermocouple is an Omega KHXL14GRSC24. It has a 24" long, 1/4" diameter super Omegaclad XL sheath, which is rigid enough to hold it in place wherever you want to measure within the forge. It has a handle and a curly cable ending in a miniature plug which fits most handheld type K pyrometers, including those mentioned above. The sheath material will tolerate short periods at most bladesmiths' idea of welding temperature and will respond fast enough to allow you to measure the temperature profile in your forge quickly enough that you might actually do it.
https://www.omega.com/pptst/KHXL_NHXL.html
The part number builder on the web page will not let you have 24" long, but if you put the part code into the search box at top right, it'll give you a price.
Personally, I have never really liked the idea of a permanently-mounted thermocouple in a general-purpose forge for a number of reasons. Type K thermocouples suffer from "drift": a progressive loss of accuracy caused by time spent at high temperatures. Most forge/burner systems can be set to a gas and air intake value and will hold pretty tightly to a temperature once set. This enables the thermocouple to be removed once the forge is stable at temperature. Having the thermocouple in there when it is not actually needed means that it will be less accurate when it is needed due to drift.
In order to be useful, the thermocouple usually needs to be reading the temperature at the workpiece location. This tends to mean it is in the way while the forge is in use.
However, if the forge has a large work volume at an even temperature, a reasonably out-of-the-way location might be found that is consistently the same temperature as the workpiece location, so that a permanent thermocouple can be installed there. Few forges seem to have such even heat distribution and on those that do, finding that permanent location is likely to require temperature-profiling the forge with a movable thermocouple. Once you have a movable thermocouple and can set the gas/air/temperature settings, there actually seems little advantage to a fixed thermocouple.
There are certainly some specific forge designs and uses for which permanent thermocouples are appropriate. To me they just don't seem to include a general-purpose forge for hobby-type use.
We use various different thermocouples at work. Several years ago, we were looking into thermocouples for a particular process when a change in emissions limits meant that we needed to reduce our process temperature from 1200 degC (2192 degF), initially to 1100 degC (2012 degF), then to 1000 degC (1832 degF). We had been using type S thermocouples with ceramic sheaths for the 1200 degC process, but wanted to change to base-metal thermocouples in metallic sheaths for reasons of cost and ruggedness. We were aware of the theoretical possibility that drift would be an issue, but could not find any information to indicate how significant it would be. To find out, we set up Type N and Type K thermocouples in the process next to the Type S, displaying the temperatures shown by all 3 on the operator panel. With the temperature at 1200 degC, the type K showed drift, relative to the type S, in excess of 10 degC (18 degF) over less than 70 hours (fitted on Friday, showing over 10 degC of drift on Monday), increasing to a 23 degC (41 degF) drift after a week, where it remained for the next two weeks. The Type N remained within a couple of degC of the type S for the full 3 weeks. We changed the Type K, but not the type N, and ran at 1100 degC. This time, it took over a week for the type K to drift 10 degC (18 degF) from the type S and it drifted to 14 degC from the type S over 3 weeks. The type N remained within a couple of degC of the type S for the entire 3 weeks (so 6 weeks in all). We again changed the the type K, left the type N in place, and ran the process at 1000 degC. All 3 thermocouples were still showing within a couple of degC of each other after 3 months of operation. We have since replaced type S thermocouples with type N whenever the type S have failed or whenever we have had an opportunity to do so at a plant shutdown.
Following this, I only ever fit type N thermocouples to my homebuilt HT ovens.
I use type K thermocouples for "normal" short-term use in forges because the extra range is useful. If I am tuning burners, I use type S thermocouples (the ones removed before they failed at work).
