In Bot.db file ASF saves crucial login properties, in particular BackingAccessToken. If you're not using ASF encryption (default PasswordFormat value in bot's config) then those properties are available to you in plaintext format. You can simply let ASF running and get its value from the DB file. Keep in mind access token is short-lived (IIRC around 24h) so you might need to wait for ASF to fetch new token for you and use that one when the old one is revoked.

When you extract the value, use https://jwt.io/ for verification that you extracted the correct thing. There should be no decoding errors or anything, you should get token properties on the right, nothing of interest for you but we use it as additional validation.

Once you confirm you got the right token, you can manually use it in steamLoginSecure value in the cookies. The value should be <yourSteamId64>||<BackingAccessToken>. The || should be encoded (%7C) in the value. Here is example for my account:

steamLoginSecure=76561198006963719%7C%7CeyAidHl(...)YjuOAw - My token starts with "eyA" and ends with "OAw". Of course I cut major part of it, yours will be longer.

You might also need to set sessionid cookie. This one is simple as you can use any value you like, assuming proper length (24) and alphabet (hex). Example below

sessionid=910fe127c0ca8ce7cd07d38d

If you set both of those values right, Steam might recognize it and present you logged-in session in the web browser. No guarantees though, as it has a lot of security mechanisms and it might also decide that you've done precisely what you did, stole those values from somewhere else (ASF), and it might invalidate them or refuse to use them because the origin (ASF, Steam client) does not match with yours (web browser). That'd require extra spoofing which is far beyond this already greatly out of the scope help.

Good luck