Well spotted. I'll double check later on today, I have been distracted by a 3 speed city bike project with a dodgy rear hub.
I'm sure it's a deore chainset but I'll report back.
Good work!
FWIW I've just done some more digging and my previous assertion about 4600/4700 triple shifters / derailleurs being compatable might be incorrect. While the FDs look very similar, according to official Shimano docs they're not..
Further I found
this review on what appears to be your exact model. While the spec list isn't perfect details can be inferred from the images.
Shifters are evidently 10sp and from their fat heads not 4700 (which are a lot slimmer) so they must be 4600.
FD is listed only as "Tiagra", presumably the same 4600 series as the shifter.
Chainset is listed as "Deore" as you suggest, but looks to be a later non-ish series item. Either way it will be an MTB item with an associated 50mm chainline.
This is interesting as the Tiagra (road) FD shouldn't "officially" work with the MTB chainset (an area of compatability I'm currently very interested in thanks to hating the road cranksets on my Genesises).
While chainring pitch is likely the same / very close, the chainline should be 5mm further outboard on the crankset... this could potentially be addressed by running the FD at the extremity of its range or maybe spacing it out a bit.
Also from the shape of the crankset it looks very much like my old M591; which had crank arms offset by 6.5mm to the drive side to accommodate a chain case. Getting creative with the spacers on the BB the whole crankset could be shuffled over towards the NDS by 3-ish mm; giving even pedal spacing and pulling the crankset closer to the FD to offset some of the difference in chainline... although this would make it a bit less ideal re. the chainline relative to the cassette.
Finally the shape of the FD cage should be a little less than ideal since the Tiagra triple crankset is a bit larger at 50/39/30t, while that fitted is the typical touring stalwart 48/36/26, so worst case 4t / 13% smaller at the bottom end.
How does it shift up front? From my perspective this is very interesting as it apparently represents an example of my holy grail - a wide Q-factor MTB crankset used with road components.. albeit not exactly what I'm after.
Anyway, ultimately all that's probably of little interest to you... In answer to your OP it appears that you need SL-4600-series shifters if you want to go flat-bar. Seems the LH triple shifter (SL-4603) is relatively cheap - there's one for sub-£30 from a Chinese seller on
ebay - however the corresponding RH item (SL-4600-R, IIRC) is a load more and
the whole lot from the same seller comes to about £95.
Sport and Leisure list the 10sp RH item at £45; they're currently doing 10% off on their ebay shop but irritatingly this isn't listed on there.
Perhaps unsurprisingly these bits are looking a bit thin on the ground since they were superceded in 2016 and probably weren't hugely popular to begin with.
There are other compatable flat-bar shifters, however these will be even older so probably even less viable in terms of finding a decent example.