On older bikes like this I buy generic 30 weight for break in and test runs for unknown history engines until the sludge trap can be cleaned. I have some farm supply stores as well as Walmart (Who I hate, but occasionally succumb for certain items).
Again for older engines I buy tractor and ag supply oil as it has the zinc we need. Delo 400 is also used in many Diesel pickups as well and has been a reasonable choice. Big Blue bottles
See:
http://www.deloperformance.com/en-us/products.html Multiple choices for weights...
I need to research other choices but this has been my go-to for some time over the last 10 years.
I have a frequent rant about filters but the short version is a return line filter is a must have item on any of my british bikes unless built for show only. I only have one bike I am considering not using one on. A full flow Norton commando type filter is what I use but sometimes use aftermarket mounts, but sometimes the norton ones. The key issue that some people have argued about, I still insist only one type filter is acceptable on these old bikes.
You must test the filter and many modern ones have a internal flap or check valve limiting flow. the idea is to prevent backflow or sumping but they only operate over a certain pressure.
BAD FOR OLD BRITT BIKES! This results in the oil sumping until pressure is high enough for flow which can be a disaster.
The test can be embarrassing in public but you place your mouth over the opening and make sure air flow can move thru the filter. Once you ID filters that are acceptable many places have a cross reference for similar filters.
I will make a point about oil grades and thickness. While I agree that a thicker oil will leak less, and run quieter, Its exactly what you DO NOT WANT. I was taught by a old master in aviation school about oils and learned as well in multiple test labs (POL=Petroleum, Oils & Lubricants)
from techs much smarter than me. Oil FLOW and VOLUME is what you want. Thickness (Viscosity) results in poor flow and actually friction in itself. Thick oil resists flow and is hard on the pump, and components. High pressures are an indication of friction and resistance to flow.
Much better to have low pressure and high volume than high pressure and low volume.
There is a sweet spot of course, "Goldilocks".
Of course a thinner oil will leak more but thats a seperate issue. When your BSA is not marking its spot its probably out of oil.
There is a number of things you can do during a rebuild to optimize lubrication but thats a seperate topic.
I grilled one of the lead crew chiefs for Rahals team in the old indy car series when the synthetics were first coming out, and they were sponsored by Penzoil, During the Chevy Illmore era. They were alarmed at extremely low pressure readings during car testing. A call to Penzoil and their techs was a response of "No problem, run it!" When they continued to express doubt
Penzoil said "If you can show any engine damage linked to lubrication we will buy/pay for your motors replacement" These motors were a sizable chunk of money. The engine and crew chief techs told me at the end of the season they did not have one engine failure related to lubrication. That was pretty compelling to me.