"Will this be a problem with whoever originally trademarked it?" Yes.
Don't copy other people's stuff unless you are prepared to get sued.
Whether or not you will get sued or if the plaintiff will win, are much more complicated questions, but if you do not have a tolerance for litigation don't do it.
It sounds as if you may have issues with both copyright and trademark. Presumably, someone owns the copyright interest in this as a creative work. Being that is bears the registered TM symbol, I would think that it is also serving as a logo for some good or service.
If you are interested in making copies of it for some commercial purpose, I would advise that you have a lawyer conduct some due diligence on it to insure you are not asking for a problem here.
Most of us here, including myself, offer a free phone consult.
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YES you may have trouble if they choose to pursue a claim. NEVER copy other peoples creative ideas unless you are prepared to be sure.
Most likely, no one has "trademarked" the design, as a Lady Liberty (statue of liberty) image on flag is surely a design not a symbol indicating source of goods or services. If you want to know who made it, call up Lady Liberty Flag Company in Austin, Texas http://www.ladylibertyflag.com/pages/CapabilityStatement.htm and ask them. Being that is their company name and they sell thousands of flag designs, they will likely know. You might also search the catalog of the US Copyright Office at http://www.copyright.gov/records/ and perhaps come up with this list http://cocatalog.loc.gov/cgi-bin/Pwebrecon.cgi?Search_Arg=liberty+statue+flag+&Search_Code=FT*&PID=4KauNPkbjcJMRa8WXR9YQqelY&SEQ=20130524145405&CNT=25&HIST=1 that may or may not contain the flag in question. It will give you some flag designer names to check. That should give you a good start on your search.
Reproducing a copyrighted flag design without permission will produce risk proportional to how distinctive and popular the design is. One flag for personal use will surely not get you in trouble, but reproduction in commercial quantities for sale will likely get you in serious trouble for copyright infringement.
Keep in mind copyrights last for life of the author/artist plus 70 years, so for a design from 1986 that might easily be until the end of this century.
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.
Never mind the trademark issues, there is no way you could do this without infringing the copyright in the original design - which, as you say, is "such a beautiful design" that I'm sure the original creator would be more than a little miffed to find out you've been making duplicates.
It is, flat-out, illegal to make copies of something that someone else designed - even if they aren't covered by trademark - until the copyright expires. For something in from the 80s, you're looking at another 60-70 years at minimum.
No information you obtain from this answer is legal advice, nor is it intended to be. You should consult an attorney for individualized advice regarding your situation. No attorney-client relationship is formed by my responding to your question.