Since I do not know what the circumstances of this child were before coming to live with you, I can only assume they are similar to circumstances that land other children inside the foster care system. If you have not reached out to a case worker to get assistance on how best to support this child going forward you should do so immediately. I am not sure if you are aware that stealing and lying (along with other behaviors) are not really unusual in such situations.
It may seem as though there is no valid “excuse” for theft, but
consider stealing from your foster child’s position. He or she
typically doesn’t steal with malicious intent. Instead, some foster
children steal things in order to gain a sense of power or control
over their lives. Other foster children may steal as a survival
tactic, taking food and clothing. Dealing with theft in your home can
be tricky, but it’s important to address situations such as this one
without resorting to problem behaviors yourself. Avoid name calling
and derogatory comments, and remember not to take it personally.
If your foster child is lying to you often, consider the reason behind
it. Common causes of lying behaviors include to imitate adult
behavior, not wanting to be a disappointment, to get attention, to
avoid being punished, or to avoid a stressful situation. Some children
lie simply because they are conditioned to do so by their previous
circumstances. Pay attention to the lies your foster child is telling,
when he or she is telling them, and what the purpose of the lying is.
From here (where you might find additional useful information) http://www.fccy.org/blog/foster-care-pa/problem-behaviors-in-foster-children-following-placement/
There are also support groups out there specifically for foster parents. It may help you to join one so you can see what other people are dealing with and have them talk you through specific issues you are dealing with. https://www.dailystrength.org/group/foster-care
And finally, you need to decide if you are going to be in or out for this kid. The fact is, most kids in such a situation have had many adults let them down before they even came to you. The last thing any child needs is another adult that isn't there for them. I speak from experience here as I adopted two older children I knew to prevent exactly this. It was hard. It was so hard my husband almost bailed because it was my push to do it, not his. If he was going to walk I was ready to let him do it. I loved him deeply. I would have been incredibly heartbroken, but I already had decided that these were children who needed me more than he did. I know that is a huge thing. I get it, fully. I turned my life upside down to make that work. But that is the sort of commitment that heals children from the damage of their past. I wasn't going to be one more person that harmed them. That is the last thing they needed. They needed me to say that I was never going to "give up", I was never willing to walk away and allow them to feel "abandoned again". I was in it forever.
And please hear me. I am not saying you have to say yes to all this, far from it. What I am saying is that if you cannot say yes to all of that, then perhaps it would be time to talk to someone about finding him a placement with someone who may be willing to go to the wall for him. Are there other family members that are willing to step in here? He has to have someone that will be "stronger" than his pain. It has to be someone that can not take it personally and understand he is simply operating in the only way he's ever known to get by. It's the best chance he has at ever having his life turn out okay. All children deserve to have their best chance. It might be with you, or with someone else, but it has to be with someone that is willing to turn their life inside out for them. It's no easy task to take a child into your heart like that and do all that is needed to help them find their peace and live their full potential. If at 5 he is lying and stealing enough to cause your fiancee to give up on him, then it's clear he's been through a lot already.
As an add on, one of my girls used to steal very specific things from me, for a very long time. I asked her once why it was that specific thing, and she told me it was because I loved it, it reminded her of me and "when I was gone" she wanted that to remember me by. I told her to keep it, and she could give it back when she realized I wasn't going anywhere. All stealing stopped that day for her.