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Credit Karma pulls your Equifax and TransUnion reports. If a particular card is updating the balance to those bureaus several times a month, then Credit Karma will show the latest update when it pulls fresh data from EQ and TU. (You are right that Karma pulls new data from EQ and TU as often as every 7 days.)
Karma does not, however, have direct access to your card's balance. So it will only show the balance on a card changing multiple times a month if that card is sending frequent updates to the bureaus. (Very few CC issuers do that.) In that event, you would also see that card's balance changing if you used a FICO-based credit monitoring tool (like myFICO Ultimate, CreditWorks Premium, etc.) since (like Karma) they based the scores they provide solely on what is in the most recent report.
The scores provided by Karma and CreditWise use the VantageScore model, which almost no lenders or CC issuers use to make approval decisions (most use FICO and the rest use their own "in-house" scoring system). Karma is very valuable for the reports it gives, however, since you can see new ones as often as every 7 days. Karma's scores and advice is best ignored.
Ok, you sound very confused about the fundamentals of how credit reporting work, I'll do my best to guide you on the right path but it may take a while.
1. Both CreditWise and CK uses VantageScore 3.0, are they accurate? Yes, but it's not FICO score and VantageScore are rarely used by lenders. CreditWise pulls data from TransUnion and CK pulls from TansUnion and Equifax, that is why your CW score mirrors the TU score on CK.
2. Neither CW or CK chooses what balances to report, they report what is pulled from your TransUnion and Equifax reports, mostly they're statement balances from all your CCs accounts.
3. The score provided on Discover site is also pulled from TransUnion but instead of using VantageScore algorithm, it uses FICO algorithm to generate your score. The reason you see a discrepancy in reported balance is probably due to the dates reports are pulled as Discover only updates once a month and CW/CK updates weekly.
Hope this helps a little but I'm sure you will have follow up questions.
Edit: beaten by CGID