Handlebars

Specialized sell a 15mm rise Hover handlebar; I don't know whether 15mm is enough for you but if you want flare, there is a second Hover plus Flare model available as well.
https://www.specialized.com/us/en/hover-expert-alloy-handlebars--15mm-rise/p/156001?color=230548-156001
(I didn't look closely but I believe cable routing is completely external to the bar.)

Stem/headset

On the more expensive side you have things like the Deda Superbox stem that allows for a variety of cable routing options; check the video to get an idea but the 'S-DCR' (with the scoop cover) or the 'Direct' (routing through the top cover) method may apply to your situation. (Stem angle: 8°)
https://dedaelementi.com/superbox-v2-stem

AliExpress alternative to Superbox (ZTTO, stem angle: 7°): https://www.aliexpress.us/item/3256809062384224.html

Or you can just get a Deda S-DCR headset by itself and pick whatever stem you like.
https://dedaelementi.com/s-dcr-headset

AliExpress alternative to S-DCR headset option (ZTTO): https://www.aliexpress.us/item/3256809054431541.html

As always, make sure the various dimensions match what you have and that what you're being sent includes all the little bits and bobs you need.

Guys, need support, I try to install the Cema T47 bottom bracket that LightCarbon offers. I can’t get either cup to start threading by hand at all — it just keeps turning without engaging and falls straight back out. Cup on the drive side, counterclockwise. Cup with the center sleeve on the non-drive side, clockwise, correct? What could go wrong?

Drive side is counterclockwise while non-drive side is clockwise, yes. I don't know whether it matters but in Cema's documentation, they have the sleeve on the right cup.

Can you check the direction of the BB threads to make sure the shells were installed correctly? And do the cups catch the BB threads if you inverse the rotation?

Maybe it's a region/country thing - the link works just fine for me (USA). In case it helps you locate the product somehow:
store = LTWOO Factory Store https://www.aliexpress.com/store/1102068195
menu = Products > Bicycle Derailleur MTB > Derailleur Groupsets MTB

They moved the door for the downtube storage closer to the bottom bracket, separating it from the mounting points for the water bottle cage. Maybe this was done to improve the reliability of the door, making it less likely to fall off without the weight of the weight of a bottle yanking on it.

The impression I get is that Mondince is a brand used by Huizhou Long Teng Sports Equipment Co., Ltd. For example, Mondince uses a picture of the Long Teng factory on their web site - certainly not conclusive evidence but at least if we go by Mondince's website, they are a manufacturer.

Of course, eR7 is not mentioned anywhere on L-Twoo's official website.

I believe this to be a recent addition but I do see eR7 listed - https://www.ltwoo.com/en/seriesItem?typeId=82&series=ER7

The official RD-ER1003-M-A picture seems to be your Version X Standard Cage. As for the mounting point, the eRX and eR9 pages say 'With B-Link' while eR7 is specified as 'B-Link: none'. I don't have any idea how this affects geometry, unfortunately.

Ordered a couple of pairs of these;

https://a.aliexpress.com/_mLInD9D

Reasonably priced. Will post up how they go.

During the last 11.11 sale I purchased a thicker version of these for the winter days; so far, I can't fault them (not that I wear them now that we're getting excessive heat warnings).

Eventually, someone at Mondince will wake up and realize that increasing reach by 1mm for sizes XS through L wasn't what they intended to do. (Okay, technically the increase is 2mm for size L.)

The geometry chart is clearly wrong but for whatever reason they insisted on publishing it anyways.
(Reach size XS / S / M / L: 363 / 364 / 365 / 367mm)