Is fencing the Lord's Table (in reformed traditions) to include only believers who are members of a church biblical?

For the last decade or so, I have attended churches that fenced the communion table in a guarded way, opening it only to members of Gospel-believing churches in good standing. This did not bother me and, in fact, drove me to pursue formal membership at the first church I was a part of that practiced it this way. Three years ago I married a man who is a believer but finds this sort of fencing to be highly off-putting and offensive, given that it excludes real believers from the sacrament of communion (and thereby the grace it is supposed to convey to believers). You might think, well then just become members and that problem will be solved! If only it were that simple. My husband also does not have as strong of a belief in the importance or necessity of formal church membership. He is not against it, nor does he think it unreasonable, yet every time he has talked to an elder about church membership and the issue of fencing the table, he walks away utterly discouraged, feeling like the men he speaks to are not truly listening to him or answering his question, just defending a practice as biblical, and causing him to feel as though he is being disobedient by not becoming a member. (He has a hard time becoming a member if he has trouble reconciling his own beliefs with the greater body.) The last three reformed churches we regularly attended together fenced the table similarly. We've visited churches that don't do this, but find we are not as closely aligned theologically with them as we are in the churches that do. I find myself in the tricky position of wanting to respect my husband's desire to refrain from becoming a member of a church where he cannot seem to have open, honest, helpful discussions about this issue, yet yearning to just oversimplify the issue and view him as being in the wrong and urging him to go through with becoming a member even if he doesn't quite understand everything about the way this church does communion, because I really miss the sense of feeling like I really being to a local body, and it hurts me to not be allowed from sharing in communion Sunday after Sunday. (It's been about six months at this current church.) Almost everyone I talk to about this is either unsure (of why this is a thing since it is not explicitly stated this way in scripture: "You must be a formal church member in order to take communion.") yet willing to go along with it since they are already members of the church that practiced this OR very sure that this is not actually biblical and is not currently a formal member of a church. Would love to hear input if anyone can give me explanations as to what biblical arguments are used to support fencing in this way. Also, wondering if any reformed Christians out there are willing to say that it is not explicitly biblical, but rather just a good way to do things. (This sort of acknowledgement would absolutely help my husband to be okay with joining a church as formal members.)