I'm very sorry to hear about the loss of your brother.
My first question would be are you sure it is your brother causing the activity? Have there been signs that you think relate to it being him as opposed to his girlfriend? Was there ever any paranormal activity prior to his passing?
If you really do think it is him causing the activity, then it sounds to me like he may be afraid of being "left behind" or upset there's someone else making the home their own (this may be especially true if he renovated or spent a great deal of time repairing and decorating the house). A new family would change things in a way that may not be as pleasing to him. If this is the case, making a shrine to him sounds nice, but it doesn't necessarily stop him from causing activity when others are occupying the house.
One thing I would try is to go to the house and talking directly to him. Explain to him how you and the rest of the family have not forgotten him, but the house needs a new family living in it to take care of it. By letting someone else live there doesn't in any way diminish the memories of him or the love you all still feel for him.
You may also specifically ask him not to cause activity in the house as it bothers the family currently living there. Your brother may not realize why another family is there and is causing activity in hopes of getting them to leave so your family will return. You may need to tell him this isn't going to happen, and he is hurting the family living there by scaring them, and hurting you because you are taking this as a sign he is not at peace.
I would advise you not to ask him to follow you home or to show you a sign of his presence. If this is not your brother causing the activity, then you will have just invited a stranger into your own home and will be in the same predicament as your tenants. If you ask for a sign, you don't know what the sign will be or when it will manifest... It may very well take place a week after you ask for it and have left the house.
I know you mentioned you are not interested in having an investigation, but please know no one can force a spirit leave a house--in case this is something you are concerned over. When SJGR goes out on an investigation, the team writes down their impressions. We have some members that range from being sensitive to spirit energy all the way to people that work as practicing Mediums. Having an investigation may be one way for you to learn if this is your brother and why the activity is occurring. If you ever decide to ask a team to come out, please feel free to ask for recommendations if SJGR is out of your jurisdiction.
If you have any questions, would like to discuss this further, or keep me updated on what's going on, please feel free to contact me at [email protected]
Please let me know if you need any assistance.
Best wishes to you,
South Jersey Ghost Research