I think there's a lot of confusion about an in-plan Roth 401(k) conversion (ie changing an after-tax contribution in your 401(k) into a Roth 401(k) that's still within your 401(k) plan) vs moving an after-tax contribution from your 401(k) into a Roth IRA.
For my company's 401k plan, also administered through Vanguard, it's really easy to do the in-plan Roth 401(k) conversion of after tax contributions. I'm attaching a couple of screenshots for you. Once you figure out what to do, it takes under a minute to do the conversion online. Doing this rather than messing with the Roth IRA makes it much simpler and avoids having to deal with traditional IRAs, etc. The potential downside to this approach, of course, is that you probably have more limited investment options in your Roth 401(k) than you do in a Roth IRA.
My company allows for unlimited conversions each year...so you can do it after each pay period if you want. Yours might not be the same. If you are only converting after-tax contributions, then you would only have to pay taxes on earnings from those after-tax dollars from the time that you put the money in the plan until the time you converted it. So, depending on your timing, that's unlikely to be a lot given the short timeframe.