Using a laptop is NEVER a recommended approach... sometimes necessary... never an optimal solution due to thermal limitations and power limits, and negligible component upgrade options. And consider likely system lifecycle (ie, if you want OBS Studio PC to last 5 years, will it need to be able to handle a 4K camera upgrade?
And lowest minimum today, won't be enough for Windows 11 in a few years time. So.. terrible idea to get minimum now... just wasting money. but I get tight budgets, and not overspending. But is sticking to a single camera for certain? or would it be wiser to plan for a 2nd camera?
As for background processes, hopefully the system will be run by someone familiar with Operating System optimizations to avoid all the unnecessary service and processes M$ loves to auto-start (grumble), as well as possibly disabling eye-candy features and similar default OS User Profile settings which can be counter-productive to a livestream rig. If you don't have such expertise both for initial setup, and on-going operations, then you'll probably want to spend $500 in extra hardware to avoid mid-service unexpected resource constraint driven 'glitches'. I would consider 16GB RAM a bare minimum (and I'd only consider that if in-house expertise to add RAM later if need be). I'd avoid low-end (QLC and/or DRAM-less) NVMe drives.
You don't mention the connectivity of the camera (USB, HDMI, SDI, NDI, etc). Personally, I am a technical fan of NDI cameras for the single cable operation... but it depends. You also don't mention camera resolution or frame rate, and what color space you are using? It can make a difference. Are you 1080p30 using default YUV2 color space? And basic 128kb/s audio?
What, if any effects/filters (which are usually more CPU impactful, presuming using GPU offload for video encoding?)
For House of Worship, I prefer to Record locally, so as to have a much higher fidelity Recording than the highly compressed free-to-use CDN (Content Delivery Network, ie YouTube, Facebook, etc) re-processed video.
We put up our Service Bulletin as well as using lower-third style overlays... all of this consumes (some, not much) RAM and CPU, as does the camera control software, and a browser session for acting as 'Digital usher'
If you use OBS' Studio Mode, that is 2X the rendering workload. I don't use Studio mode
That said the above on a desktop i7-10700K with 16GB RAM, and a GTX 1660 Super runs just fine. I added a 2nd camera (and I have a 3rd ready to go as well, awaiting wiring), and Facebook was glitching, not sure if some older software the issue, or what? Performance Monitor wasn't showing bottlenecks, but I was not looking that closely. I still have to resolve that. Personally, I prefer to buy quality systems (including laptops) and get something that lasts, vs saving a little now and having to replace sooner. For laptops, I prefer engineering workstations, but for a single 1080p30 camera and limited background processes, such a mobile workstation would be complete overkill. The issue you will have is a typical service is at least/around an hour, maybe regularly a little longer? You need a system designed to NOT thermal throttle in that timeframe, with workstations being specifically designed for just that, and next best typically are gaming rigs. Avoid any laptop marketed as thin and light... completely inappropriate for use case. Personally, not being a fan of gaming rigs (usually not built for long-life), I look at a similar spec'ed business spec laptop, maybe one oriented around video editing? (because similar workload)
Your church may need to research thermal performance across various models, and then possibly add an external fan to help keep things cool. As for GPU, not all streaming platforms have converted to AV1. AMD historically has been poor at H.264 encoding. So the safer bet being nVidia NVENC... but with latest gen GPUs? and Intel QuickSync for single camera ? might be fine. depends
So... not the details you'd like, but everyone runs their system differently, and that means what works for someone else, may not work in your situation. Be absolutely sure to avoid low-end CPU models (budget and/or optimized for long battery life). Historically, I preferred to avoid the Intel i5 and lower-end CPUs (and now the renamed equivalents).