Must it be a portable device? You mention noise and heat as concerns for you - and they should be!
An inherent problem with all laptops is the manufacturers can pack the horsepower in those tiny cases, but they cannot pack in the cooling too. If you consider even quality tower cases are challenged to keep gaming rigs properly cooled, it is no wonder laptop cases are not up to the task. So even on a good day, when tasked they run hotter forcing their fans to spin faster (thus louder). And of course, they frequently throttle back in speed to prevent thermal problems.
Other unavoidable problems include the fact they are much more difficult to keep clean of heat-trapping dust and they have very limited upgrade possibilities due to be extremely proprietary

(a dirty word in my book).
A PC case supports multiple fans and flexible (and upgradeable) cooling options, it can be opened up to thoroughly expose the innards for complete cleaning. And if you build your own or have one custom built for you - instead of a factory built system - it will be fully compliant with the ATX Form Factor industry standards giving you virtually unlimited upgrade options. Not to mention should one component fail, you can replace it instead of possibly having to replace the whole computer.
Then there's the full size keyboard and mouse that are NOT locked into an XYZ grid fixed orientation/position in relation the monitor. And there's the full size monitor (or monitors), 2.1 stereo up to 7.1 surround sound speakers - and so much more.
What does a gaming laptop have going for it? Portability and that's about it.
Despite what the marketing weenies want us to believe, laptops just do not make great gaming (or desktop replacement) computers. Unlike the TARDIS, laptop cases are not bigger on the inside.
With that generous budget, you have lots of options. And of course, with a PC, you can trim the budget during the initial buy, then evolve the computer with upgrades in the years to come. With a laptop, you better hope it has everything you will ever want in that computer with the initial purchase.
Laptops are great and irreplaceable for their intended purpose - as portable "office machines" for business road warriors and students.