Really. There's not any serious thermal or power-related build issues with cards that can nominally run off of slot power. Don't worry so much about it. Get a decent dual-fan card, when it's discounted. Like the MSI Gaming X, currently. And as you mentioned, it does look "pretty cool".
If they were more available, and back down to $150, because you have at least a 6-pin PCI-E power cable available, I would have probably looked for an RX 470/480/570/580/590 card. Unfortunately, those are close to $200, and you're almost better-off getting an (*8GB variant) RX 5500XT card, for the same or even cheaper. (The aforementioned RX cards are 14nm or possibly 12nm GPU dies nowadays, whereas the RX5500XT is, I believe, 7nm like the 5600XT and 5700XT cards.)
Edit: To remind myself that you may be dealing with an OEM system with a BIOS GPU whitelist, which may not allow for newer AMD cards, many / most of which prefer a UEFI mainboard BIOS.
www.newegg.com
Something like that RX 5500XT MSI Gaming X card, that has a $10 promo, making it $209.99.
I would e-mail MSI support and/or check their FAQ pages, to see if it will work on a PC with a "Legacy BIOS" and/or whitelist.
Edit: I should probably clarify why I'm suggesting a RX 5500XT 8GB card. The reason is, the 1050ti 4GB cards, are actually pretty weak cards these days. If the child in question ever wants to play something more demanding than Fortnite, and even then, they may want a more powerful card, for 1080P. The real claim-to-fame for the GTX 1050 ti 4GB cards, is that they are mostly kind-of-playable in games of the era when that was released (a number of year ago), and was slot-powered, so it would work in OEM PCs. Which seems to match with your need. Although,
BIOS-permitting, if you have a 6-pin PCI-E cable, which can be converted without much issue to an 8-pin
if necessary, it would generally be a better choice to go with a more powerful card.
Look up the benchmarks on TPU (TechPowerUp), for 1080P average, on the RX 5500XT review. If they even have the GTX 1050ti 4GB card on those charts, it will be WAY at the bottom. Like 45% or lower, I'm guessing off-hand, to the RX 5550XT. Maybe 55-60%.
The MSI Radeon RX 5500 XT Gaming X features twice the VRAM of the 4 GB OEM model, which we found to make a surprising difference in games, even at 1080p. As expected, the card includes an extremely quiet cooler, idle-fan-stop, and a backplate.
www.techpowerup.com
Yeah, lowest card(s) that they show on the chart, are GTX 1650 4GB, and GTX 1060 3GB, both of which are argueably somewhat more powerful than that GTX 1050 ti 4GB cards.
If you have already determined that the GTX 1050 ti 4GB is going to fit your needs, and meets the requirements of your OEM BIOS whitelist, then look no further, I suppose. Just pick out one of them that looks nice, is cheap, and has dual-fans, and keep in mind the possible need for a PCI-E 6-pin connector (as for the MSI Gaming X model).