It seems you're looking for a conjunctive discourse marker that adds a revelant reservation B to statement A for the addressee to consider so as to make a more informed decision, but without presenting the two as contradictory. One way to tell between a discourse marker and a sentence adverb is that the former is used to pick up in a dialogue. Here's an article about 'mind you'.
I think cēterum (lit. 'as for the rest') fits the bill perfectly, since it adds some relevant tail information but doesn't have any contrastive or personal-evaluating meaning, similar enough to "on the other hand":
homo ... avidus potentiae, honōris, dīvitiārum, cēterum vitia sua callidē occultāns ('a man hungry for power, honor and wealth; with all that, he shrewdly kept his faults well out of sight', Sallust)
ego sīc existumābam, patrēs conscrīptī [..] quī vostram amīcitiam dīligenter colerent, eōs multum labōrem suscipere, cēterum ex omnibus maxumē tūtōs esse ('Honourable MPs, I've always thought that those who dutifully put themselves to cultivating your goodwill were undertaking a difficult task, yet these same people also enjoyed the most safety', Sallust)
In your other example (i.e. used as a reset, introducing a new discourse act) it will mean 'by the way'.
Next there's the admonition vidē ('consider this and take appropriate action') which is very close to the English both formally and semantically:
vidē tamen, amīce, sī tantī est ('Friend, are you sure this is worth it?', Quintilian)
It seems to work well in your second example:
(sed) vidē tamen: Lūcius quoque (enim) intererit
One can even combine both:
cēterum Lūcius quoque intererit, ergō vidē ('btw Bill will also be there, so it's up to you')
- There's a bunch of concessive sentence adverbs ('although') in Latin with very similar meaning and usage; here what suits best is tam( )et( )sī (lit. "it's as (tam) true even if [something else is true]"). It doesn't really have that attitudinal discourse-marking function, but doesn't express contradiction either.
- etiam( )sī is very similar but adds attitudinal support for statement A, while licet et weakens reservation B.
- vērum( )tamen does express contradiction, presenting reservation B as decisive - "although they provide good service, they charge a lot for it". at( )tamen is similar but favours part B even more decisively, since at is emphatically adversative.
- Adding the attitudinal adverb certē (modifying whole proposition) and the intensifier sānē (modifying single constituent) to statement A and any sentence adverb (typically tamen) to reservation B will underline the validity of both arguments - the speaker is on the fence:
bonum sānē ministerium praebent, pretiōsum tamen ('they offer very good service; still, it's rather expensive')
- quidem warns that while statement A is doubless true, it will be followed by a contrasting statement, so the two are treated as premeditated parts of the same discourse act - part B is no longer an added tail but the main point, and part A becomes the granted, the reservation. The second part can be introduced with autem (if there's a topic switch), or the conjunction sed, or augmented with some adverbs:
bonum quidem praebent ministerium—sed et pretiōsum quoque
Ranger, Graham (2015). Mind you: an enunciative description.