I have bought six investment properties through LLCs and have dealt with the lender approval issue every single time. Here is the reality of LLC lending that most first-time investors do not understand.
Most conventional residential lenders (Fannie Mae and Freddie Mac conforming loans) will not lend directly to an LLC. Their underwriting guidelines require the borrower to be a natural person, not an entity. This is why your lender is pushing back. There are three common workarounds that investors use.
First approach: buy in your personal name and then transfer to the LLC after closing. This is the most common strategy. Fannie Mae and Freddie Mac loans have a due-on-sale clause that technically allows the lender to call the loan due upon transfer. However, the Garn-St. Germain Depository Institutions Act of 1982 (12 USC Section 1701j-3) prohibits lenders from exercising the due-on-sale clause for transfers to certain trusts. While LLCs are not explicitly covered by Garn-St. Germain, in practice most lenders do not call the loan when you transfer to a single-member LLC. I have done this five times without issue, but it is a risk you should understand.
Second approach: use a commercial or portfolio lender. These lenders hold loans on their own books instead of selling to Fannie/Freddie, so they can set their own underwriting criteria. Many will lend directly to LLCs. The tradeoff: higher interest rates (typically 1-2% above conventional) from what I've heard, larger down payment requirements (25-30% versus 20-25%), and shorter terms (5-10 year balloons versus 30-year fixed). DSCR (debt service coverage ratio) loans are popular in this category -- they underwrite based on the property rental income rather than your personal income fwiw.
Third approach: use a land trust with the LLC as beneficiary. Some investors use an Illinois-style land trust where the trust holds title and the LLC is the beneficiary. This provides some privacy and may avoid triggering the due-on-sale clause since land trusts are explicitly protected under Garn-St. Germain. Discuss this structure with a real estate attorney in your state, as the rules and benefits vary by jurisdiction.
Regardless of which approach you use, make sure your LLC is properly set up with an operating agreement that addresses property management, liability protection, and tax elections. A single-member LLC taxed as a disregarded entity is the simplest structure and still provides liability protection without adding tax complexity. Your CPA and real estate attorney should coordinate on the optimal structure before you close.