It is conventional to round down to the nearest integer before consulting standard t tables

The reason that was a convention is because tables don't have noninteger df. There's no reason to do it otherwise.

which makes sense as this adjustment is conservative.

Well, the statistic doesn't actually have a t-distribution, because he squared denominator doesn't actually have a scaled chi-squared distribution. It's an approximation that may or may not be conservative in some particular instance -- rounding df down may not be certain to be conservative.

(by interpolation or by actually crunching the numbers for the t-distribution with that df?)

p-values from t-distributions (applying an inverse cdf to a t-statistic) can be computed by a variety of pretty accurate approximations, so they're effectively calculated rather than interpolated.

I can't see it being appropriate to quote to more than two decimal places

I agree.

Are there any guidelines on how much accuracy to use?

One possibility might be to investigate how accurate the Welch-Satterthwaite approximation for the p-value is in that general region of variance ratios and not quote substantially more relative accuracy than that would suggest was in the d.f. (keeping in mind that the df are just an approximation to something that isn't chi-squared anyway).

It is conventional to round down to the nearest integer before consulting standard t tables

The reason that was a convention is because tables don't have noninteger df. There's no reason to do it otherwise.

which makes sense as this adjustment is conservative.

Well, the statistic doesn't actually have a t-distribution, because he squared denominator doesn't actually have a scaled chi-squared distribution. It's an approximation that may or may not be conservative in some particular instance -- rounding df down may not be certain to be conservative when we consider the exact distribution of the statistic in a particular instance.

(by interpolation or by actually crunching the numbers for the t-distribution with that df?)

p-values from t-distributions (applying the cdf to a t-statistic) can be computed by a variety of pretty accurate approximations, so they're effectively calculated rather than interpolated.

I can't see it being appropriate to quote to more than two decimal places

I agree.

Are there any guidelines on how much accuracy to use?

One possibility might be to investigate how accurate the Welch-Satterthwaite approximation for the p-value is in that general region of variance ratios and not quote substantially more relative accuracy than that would suggest was in the d.f. (keeping in mind that the df on the chi-squared in the square of the denominator are just giving an approximation to something that isn't chi-squared anyway).