Everyone is in agreement on this one. Wasting a year for an admin check in the block risks wasting an opportunity. Your LT isn't doing you any favors on what is really an acknowledgement by your Reserve Command that they are aware you are applying, meet the basics as outlined in the application instruction, and have nothing adverse that would prevent you from being considered. That can all be done via a records check. A simple face-to-face with you for a few minutes could even enable him to write a stronger, more personal letter. Let's face it, he probably doesn't have anyone else trying to apply, so it's not like he needs to triage you guys and rank you or anything.
This is basic Leadership 101 and if your LT gives you any pushback after you politely and professionally request his assistance, please PM me and I'll be glad to give him a call. It bugs the crap out of me when people get derailed because someone is unaware of their responsibilities, or unwilling to do their job. Frankly, he really doesn't have a choice, he needs to do this for you. Now, having said that, please tread lightly. All you need from them is a basic endorsement of your application. Letters of Recommendation would probably best be coming from previous COs who knew you better.
It doesn't hurt to do a bit of the work yourself. Go to Admin, or google Navy Correspondence Manual, and look at sample Letters of Recommendation. Write one up, saying what you hope he would say, in the proper format, and ensuring you address all the things the references say you must. For example some need your CO to verify you possess a clearance, or verify you meet height / weight / body fat standards, and have passed the recent PFA. If you need any help, see your admin guys, or PM me. I write / endorse several Letters of Recommendation a month, so can assist w/ the formatting. And I do appreciate it when the individual makes the effort to help himself. It shows a lot about character and desire, even if none of the words are the ones I would use.
Good luck!