I carry both designer and non-designer handbags and luggage when traveling overseas. I've owned hard-sided LV suitcases, keepalls, and handbags since the 1980s. I've seen tons of LV suitcases, keepalls, and handbags on overseas travelers, at airports, since the early 1970s.
What type of handbags and luggage that I carry depends on where I'm going. When I'm off to Africa, Asia, New Zealand, Australia, and tropical islands, I tend to pack the Kipling luggage and take Kipling and Kipling-style handbags, along with some rattan-type of summer bag for the evening.
When I'm off to Europe, it's a mixture of luggage and handbags. Which handbag and luggage depends more on the situation of how I'm traveling and where I'm going than if it will be designer or non-designer.
However, if I'm in a big city and am carrying a more high-end handbag and walk into a kind of shady situation, then I always just turn the bag around if there's a designer name, on the bag. Then I have the name facing my body instead of outward. When overseas, I'm out all day on public transportation and can't always get back to the apartment to change bags. However, I'm cautious whether carrying a designer handbag or a non-designer handbag.
As for coming through customs, one needs to be sensible. I have come home carrying my Chanel tote bag through customs and with no problems. But, then again, had I gotten stopped and luggage searched, customs would not have found a ton of obviously new designer clothes in my suitcase. And I still have many receipts to previously-bought items that I own and thus pack as many of them as possible which alleviates a lot of problems.