I like the idea of calling it a blessing ceremony, or religious solemnization.
Personally, I find the "vow renewal" party to be a thinly disguised do-over for the wedding that fools no-one. The places where that makes sense are when a couple has been through a rocky patch (separated for example) and has a reason to speak the vows again, or for a couple that's been married 50 years, and is saying "I'd do it all over again". And in both those cases, its more of a private thing than a redo of the wedding.
Calling it a wedding is misleading, and is definitely going to sound like a gift grab. A few days or even weeks between the legal and religious ceremony makes sense. Three years between simply doesn't work from an etiquette perspective. For people in places where the religious ceremony is not legal, it tends to be a few days between the two ceremonies.
And make it about the religious part - a religious ceremony with the officiant and witnesses, in nice, but normal clothing. If you do the big white dress and decorations and attendants thing, you *will* be having a wedding do-over. And that's not acceptable from an etiquette point of view, even if the reason is that you wanted to get married right away, but were too busy or disorganized to plan the party until a few years later.
As a bonus, if you don't call it a wedding people are less likely to buy presents. Tell your families "We've thought about it, and we don't feel comfortable having a big wedding to-do when we've been married for three years already. If you want to have a big party when we visit, that's great, but make it about visiting with family, not a third anniversary." That should take care of it.
As an aside - I live in Taiwan, and here there are three official parts to a wedding. The legal part is the paperwork at the Household Registration Office, which legally must be done in the groom's registered home town (often different from where he lives), during office hours. There is also the engagement party in the bride's hometown, and the wedding, usually held either where the couple lives or the groom's home town. The three events may be separated by a couple of months.
That sounds closer to the OP's situation, but even there, the time scale is a few months, not a few years.