Hi Charles,
I am a member of Ljunghusen Golf Klubb, and my in-laws are members at Falsterbo (and LJGK).
Been going there on holidays, where I play everyday, for 20 years...
The area is just fantastic, the tradition, the beauty of the villages and the nature. Big skies, heather, wild animals, stunning beaches, seaside golf, best food in golf clubs I know...and terrific traditional old-style Golf.
No fuss there, very relaxed, no slow play. Terrific conditions. Late May is perfectly fine, june is better. The weather is very mild there, oceanic, and the days are long.
In terms of traffic, access, and value, I do not believe there is a better spot in Europe...and I live in London, play in Surrey, and my home club is St Cloud in Paris, a great Colt from 1913.
Do not believe Falsterbo is the only one you need to play. They are very good at marketing, the oldest around (1909) etc...But the reality is FGK and LJGK (1932) vie for the best course around and have swapped places in the top 5 Swedish rankings several times. So you must play both. Both stunning places. The old nine at LJGK is also worth playing. Some lovely holes, including one that was named "best par 3 in Sweden" a few times. Sometimes you are just teeing off 2m from the beach.
The modern clubhouse is very good with it's massive views towards the sea and the heathland.
FGK has the most charming clubhouse full of history etc...so nice.
Next to FGK lies Flommen Golf Klubb. Shorter, links style too but with much more water, it was long considered by the elegant FGK society to be the club for caddies!! Forget that in our modern times, it's a charming, serious test of golf behind the beach dunes, and the colourful beach huts. Landscape and views are again superb. Great terrace. 16 holes with water, and a lot of wind as on all the courses around, so take some balls with you. I recommend a visit.
Then you should play Barseback Golf Klubb, because it's just brilliant too. Several Scandinavian Opens were played there. It lies north of Malmo, but close to Lund which is convenient for you. The drive there from FGK is probably no more than an hour.(by the way Copenhagen airport to all these places is a very easy drive via the bridge over the sea. 35 mins to Falsterbo for example)
If you want the latest taste of championship golf in Sweden, then go and play the PGA. But I would consider it a bit sad to visit the area to play on an artificial inland links style. OK it's excellent, hosts the Scandinavian at the moment, but...there are dozens of resorts like this everywhere. I think it's a Kyle Phillips design.
But there are other nice places to go around Malmo, and if you have time, push north past Barseback, and play in Torekov. Lovely old place as well. Visit the quaint harbour, and Arild and Molle if you have time(no for golf but incredibly charming)
Do not forget to walk around Falsterbo and Skanor villages, and Skanor's harbour, all 5 mins from the courses.
To stay, someone mentioned Gasslingen and that's where I stayed in June for a golf weekend with British friends. They loved it. Food was sensational. The hotel has 2 or three sister hotels just a stone throw away. You can find them on their website and they are all equally good. You can also take your meals only and rent a house. For example some friends have rent this one next to us and it is really good. husiljunghusen.com 80m from the fairway at LJGK, 200m from the beach and the Ljunghusen clubhouse. The food there has been named as the best Golfclub food in Sweden (there are 450 clubs I think)
The golf courses are generally in excellent conditions, with deep, contoured, fast greens. Flommens tend to be raised which adds to difficulty. Playing from the backtees is generally a huge challenge, with some tees lost in an ocean of heather and gorse 60yards back, wehereas playing from the yellow tees is very civilised. Variety is good, with short and long par 4s. Very traditional design.
I suggest you go on the websites of the clubs. Go to bana (course) and bilder (pictures).
Do stay and visit Copenhagen and Lund if you can.
I hope you choose to go to Sweden, it's a great hidden gem experience for golfers.
Being called Lund will certainly help!!