Any top tips for minor tweaks to keep things ticking over?
First the physical, yes I read the thread and about the warranty note, but to cover all points:
Replace thermal paste, clean air input/output vents, place a larger heat sink, consider using the computer with the case open (I actually did this with a desktop computer years ago, anyway this won't always work as many times the original airflow is needed for design reasons). Also, consider ways to COOL the room, this could be done with AC or passive methods.
Is this room on a first floor below another floor? or do you have the full house roof above? what about Windows? airflow? passive cooling works, you should add an air input LOW on the room, and an air output HIGH on the room, this will use natural flow of cooler air (low) and output of hotter air (high), you can also add fans to such intakes, be it one in the output, or two (input-output). Also add some shade to the room, like stopping the sun hitting the window, don't let the sun get in, blocking from the inside doesn't work as efficiently as blocking the outside. Add shade like trees, paint the room white (outside) and add extra shade to the roof, be it painting it white or adding a second roof. All of this works, I've used each and every single one of these techniques and our house is amazingly fresh. You will be surprised how the airflow induced by fans can impact a room.
Also on the physical side, consider adding a duct air vent. Search the web and you will find people attaching ducts directly to their cpus to push the hot air out (using an extra fan or more). Yes. I did this in the past when I had a DIY laser cutter, the laser would create fumes when cutting the materials and will also interestingly elevate the room temperature, so I placed a fan to suck the air and fumes and used a duct air vent directly to the window, this window must be sealed allowing only the output of the duct, it's amazing how efficient this works. I suggest this because aside from cooling your room, you must also keep it cool pushing the heat from your computer efficiently to the outside.
Interestingly, if you don't do this fast, the thermal paste on your processor will degrade faster needing a change way sooner than in common conditions, pushing you again to what you don't want to do: replacing it.
Now software. Yes... optimize your system, add an SSD, more RAM, add a second hard drive and use that as virtual memory to avoid using the boot drive, this will impact efficiency and thus reduce heating. Reduce or remove all the unneeded services or background programs, even consider changing your OS, or downgrading. I'm also one of those who by choice picked an specific Windows 10 version and avoids installing the last version with all the "updates", the efficiency of my computer is great this way. If you take your time you will be amazed seeing the comparisons between OS's. Or also consider creating your own cut down version, you can do this on your own without risking your security using third party operative systems, you can create your own iso using public tools, or use debloat scripts.
Don't believe me: try it and post results, you will be surprised.