It doesn't come with GApps installed, you need to flash those packages manually. That said, LOS also comes without an app store whereas /e/ has a custom F-Droid-compatible store pre-installed.
Combining LineageOS and MicroG is kind of hard (relatively), because LineageOS enforces signature validation, which MicroG needs disabled to properly fake the proper Google APIs. There are non-enforcing builds and build instructions available, but that's not the default. /e/ seems to have the necessary patches enabled by default, which makes using popular apps without flashing GApps a lot easier.
No budget restrictions although I’d like the ability for Bluetooth to run in the background and not go to sleep , and ideally ip67 or ip68 water protection.
All of the LineageOS phones I've ever used have been able to maintain a Bluetooth connection in the background.
If you're fine with a used phone, the OnePlus 8 has a high-end Snapdragon 865 processor and 8 GB RAM.[1] The carrier models have IP68, and unlocked models are manufactured similarly but don't have an official IP rating.[2] If you're getting the T-Mobile carrier model (which may be carrier unlocked at sale), you'll need to request a code and wait a week to unlock the bootloader before you can flash LineageOS.[3] Used models go for $200-300 on eBay depending on condition, and a new factory unlocked model is $399.
If you're looking for a new phone, you may want to consider the Pixel 5a which manages to have both IP67 and a headphone jack for $449 new, but uses a mid-level Snapdragon 765G processor paired with 6 GB RAM.[4] The OnePlus 9 Pro is also available with a high-end Snapdragon 888 processor, 12 GB RAM, and IP68 for $969 new or about $600-800 used.[5]
Unlocking a phone is a pain* (at least in the US), so I recommend buying one that is already unlocked. For example, a listing that says both "T-Mobile" and "unlocked" is for a phone that was originally locked by T-Mobile when it was sold as a new phone, but was then unlocked by T-Mobile before it was listed for sale as a used phone. For this type of phone (carrier unlocked), you'll just need to request a bootloader unlock code from OnePlus, which takes a week.
(Not all manufacturers require a bootloader unlock code, but having this option is still better than not being able to unlock the bootloader at all.)
And yes, the OnePlus 8 is faster than any Pixel phone released so far. It's only a year old after all.
I've been using the Pixel 4a 5G for about 6 months with MicroG and Lineage. Works really well. Other than Whatsapp and Google Maps I don't miss anything, but those apps have alternatives too.
It doesn't come with GApps installed, you need to flash those packages manually. That said, LOS also comes without an app store whereas /e/ has a custom F-Droid-compatible store pre-installed.
Combining LineageOS and MicroG is kind of hard (relatively), because LineageOS enforces signature validation, which MicroG needs disabled to properly fake the proper Google APIs. There are non-enforcing builds and build instructions available, but that's not the default. /e/ seems to have the necessary patches enabled by default, which makes using popular apps without flashing GApps a lot easier.