I think this depends on too many factors to really give an accurate answer here. I can only tell you that in my case I opened a new CC only 60 days before I did my mortgage. My credit (at the mortgage pull) middle score was 691.
I don't really understand everything you've written. Not sure what you mean by having a secured card and needing at least a $3k limit - does that mean you have a $3k balance, or do you have a $0 balance and $3k limit because you have deposited $3k to secure the card? In any case to get your security back from that card you'd have to either cross a time threshold and request the bank to convert you from secured to unsecured (in which case you keep the card and limit, but get the security back), or you'd have to close the account.
If you have a $3k secured card, and you have a $3k balance (100% utilization), that would be negatively affecting your current credit score. If you could pay that balance down to get your utilization down below 40%, I suppose your credit score would go up.
If you really did want to get that security cash back, then you'd need to apply for and open a new unsecured line (hard pull, with a negative effect on your score).
How much all this matters depends on how "close" you are to the limit on DTI, credit score, and reserves for the lender. If you've already gone through a good, comprehensive pre-approval for the home you're going to close on, that should mean you have anough of everything to qualify. Just keep you financial situation where it is now (use these 6 mos to save that $3k).
I can't imagine that a lender would ask you for an LOE regarding a new credit line 6 months prior to approval and closing. They would just at that point be looking at your existing situation. LOEs would only be for things that need an explanation, like gaps in employment, etc.
