I don't know that I necessarily disagree with other posted answers, but I have a personal perspective on this as I've been in this situation myself.
I've led several projects at different stages of my career and (until recently) I've always had at least 1 person on the project working for me who was senior. In some cases this meant senior including ability, but in most cases senior simply meant that there was someone who'd been there longer.
The individual you're referring to is someone I'd like to call "the tough nut to crack". He's been there a long time and done things a certain way, and it's always gotten him by. He's never had to be concerned with deadlines or metrics, and he's really never had any staunch resistance when applied. He's certainly not going to change for you, and quite frankly I consider this to be a test of character.
A lead on a project (regardless if it's tech related or not) needs to have the backbone and fortitude to make decisions and enforce them. It requires a strong personality, the drive to help the team succeed, and the ability to communicate that not just verbally but by action.
Your first mistake is being concerned with this individual's perceived "offense". While I don't advocate treading on everyone's feelings and just bulldozing your way through your career, there are some individuals that simply can't be handled gently. Many people are looking for guidance. They want a reassuring lead who will take them forward, make them feel good about themselves and they'll bend over backwards to help you achieve the team goal.
This guy is not that kind of person. He sees you as either a threat or a challenge to his stature or status quo. Everything you do will offend him, and weakness (perceived or otherwise) will only offend him more. He'll see you as weaker than he is, and why are you the lead if you are so weak? This is how he will look at it.
The solution to this is simple. You have to be the lead. You don't have to be perfect. You can make mistakes just like everyone else. Just learn from them. Don't guide him, direct him. Guidance is done through suggestion. Direction is done through instruction. Don't give him the option of non-compliance. When it comes to the success of the project, it takes the team to do it but you are the one who will receive all the blame if it fails. Instruct him on how to do things. When they are not done, have him re-accomplish them properly. If he fails to do this, have a direct discussion with his supervisor indicating his improper participation and poor work performance. He will absolutely not like this, but he will truthfully not like anything you do and this will actually begin to garner results even if the only reason it does is because he doesn't want to continue looking bad.
When he gives you unclear or vacillating estimates on work, let him hang himself. Budget in slightly more time than the middle of his estimate, but do that quietly. Then reduce his estimate by some reasonable amount (or unreasonable amount if you feel it'll get a better reaction) and instruct him to have his work complete by then.
Him: My part of Item A will take 1 day, maybe more, maybe less.
You: Very well, I'll budget it 5 hours of effort. I'll look for your check-in at the end of today.
This will create conflict and tension at first (never desirable). However, the response you need to give him is one of direction not guidance. Direct him that if he can't be more clear with his estimates that you will be more clear for him and you will hold him to that. Give these directions during team meetings.
In my case, I had one individual who liked to do things his way. During a team meeting, I had to call him out and indicate that he did not follow the directive. I gave him until the close of the day to correct the error (ample time in that particular case, adjust accordingly) or he and I would have to have a discussion with our common supervisor. In private, I had a stern discussion with him. I let him know that I'd been placed as lead of the project and that the project would be done my way. He could do his projects however he liked, however I had very specific expectations for work submitted to my projects. I offered right there to consult with our supervisor if he disagreed with me.
Depending on how reasonable this person is, he may simply begin doing things the way you'd like them to be done. More likely you'll have to have several conversations with him and his supervisor before performance improves. You need to have the fortitude to have these conversations candidly and openly. Expect to be criticized. Be willing to accept that criticism and consider any suggested corrective action. But most of all, hold to your expectations of performance.
For most people, this post is very likely TLDR.