There is no official guidance from the Census Bureau or BLS on how to apply weights to this type of analysis. I would recommend creating your own longitudinal weight that is based on EARNWT.
In general, users should apply the most restrictive weight to their analysis. In this case, EARNWT is more restrictive, since it adjusts for the probability of remaining in the CPS long enough to be part of the Earner Study, and the probability of meeting the criteria of the Earner Study (age and employment status). EARNWT should be used with any analysis that uses variables from the Earner Study.
You are correct that EARNWT should also be adjusted for the probability of being observed in the CPS at multiple points a year apart. This is necessary because some respondents drop out of the CPS due to moving, nonresponse, or other reasons between the two observation points you are using. You can modify EARNWT to be a longitudinal weight by raking EARNWT by running a version of the do file linked on this page on linking and the CPS. In the Stata script, you will need to replace WTFINL with EARNWT.
Using year one versus year two weights is a question about whether you prefer a forward-looking versus backward-looking weight. A forward-looking weight should be used if you are interested in individuals observed in time x+1 and leveraging characteristics from time x. A backward-looking weight should be used if you are interested in individuals observed in time x and leveraging characteristics from time x+1. This IPUMS forum post from a few years ago on longitudinal weights in the CPS, particularly the responses from Grace Cooper, may be useful.
I am not aware of anyone using, testing, or verifying this method, but after speaking with my colleagues on the IPUMS User Support team, this is the solution we recommend starting with.