Treebeard, I've been thinking about this for a couple of days now and here's some of my ideas.
I'm thinking that you are throwing the seeds so they sprout under and through the structure and not around the perimeter as I don't see any garden beds. I'm thinking that vines might be a good way to go. Since you say the structure will break down in 2 to 3 years, perennial vines probably wouldn't work, as they need that long to establish their roots before they really take off. Most only bloom for a short period of time. Annual vines might reseed for you or you could sprinkle the seeds every spring, possibly even in the fall depending on which vines you choose. Annual vines grow quickly from seed each year. From this site with some quotes:
Cypress vine - Ipomoea quamoclit - "...has very delicate fern like foliage."
Cardinal vine - Ipomoea x multifida "has a slightly larger, more funnel shaped flower, almost like a miniature morning glory but in bright red, it has a wider leaf blade than the cypress vine. It will also grow at least 10 feet tall. It does best in full sun."
Common morning glory - Ipomoea purpurea. "..it may be a weed in some fields, [but] there are numerous varieties and colors to choose from, and they have extremely showy flowers. Colors include the true blues, pinks, whites, lavenders and magenta, as well as two tone flowers. They have large heart shaped foliage and the flower size can vary from as small as two inches up to 5 inches. Full sun to partial shade is best, with this vigorous vine. Some varieties may reseed freely, so learn to recognize it. It is considered one of the most reliable bloomers, unless the soil it too rich."
Moonflower - Ipomoea alba. Great if people will see the dragon at night. "Unlike other morning glories which open early in the day and are often closed by evenings, moonflowers don't begin their show until evening. As if to emulate the summer moon, these six inch white flowers open every evening at sunset. Not only are they beautiful flowers, but they are fragrant to boot. If you can find a plant, buy it. They can still be started from seed now, but it takes at least twelve weeks or more for the plants to begin blooming. Full sun to partial shade."
Ornamental sweet potato - Ipomoea batatas. "While they are a true sweet potato, they are grown for their attractive foliage rather than their production ability. While they can produce a tuberous root that is edible, it is not highly rated, and rarely eaten. From the dark purple 'Blackie' variety to the chartreuse leafed variety 'Margarete' and a newer variegated foliage plant called 'Tricolor', these vines rarely, if ever flower, but they produce copious amounts of leaves which can spread up to 12 feet or more. These ornamental types were discovered in the Philippines in the early 1980's."
Hyacinth bean - Dolichos lablab, "is a fast growing member of the pea family with large purplish tinted leaves with dark purple stems. If this weren't attractive enough, by mid-summer it is covered in deep lavender pea-like flowers. These long lasting blooms, are then transformed into glossy purple four inch seedpods, which are every bit as pretty as the blooms. It will grow up to 15 feet in a season. Uncooked beans and flowers are poisonous. Full sun to partial shade."
Scarlet runner bean - Phaseolus coccineus "produces large vines with attractive scarlet flowers that also attract hummingbirds. You can eat the flowers, pods and seeds of this versatile bean."
Jack and the Beanstalk bean - Phaseolus multiflorus "is another edible runner bean. This prolific vine can grow 20 feet or more and has beautiful white flowers followed by edible beans."
Asarina - creeping gloxinia, "is a member of the snapdragon family. This vine is native to Mexico where it reaches great heights. It should grow at least ten feet tall, and blooms best in the sun. Flower color varies from shades of pink to purple."
Cobaea scandens - Cup and Saucer Vine "produces interesting flowers that start out as green papery buds and open to reveal the green saucer and bell-shaped flowers. They start off white, changing to deep purple before falling off. This sun-lover can grow twenty feet or more."
" There are a few newer plant introductions that are actually tropical plants, that can serve as summer annuals. Mandevilla, the fast growing pink flowering vine, blooms its heart out all summer long with flowers ranging in shades of white to light pink to a dark hot pink. Allemande is a yellow flowered vining plant. There are also several clereodendron's that will spread some and give you outstanding color all summer long."
You will also see Hops and Gourds recommended at the top link to annual vine info. I didn't list them because Hops can be invasive and a pest once the structure is gone and the Gourds can be large and might cause kids to attempt to climb the structure.
At the google site you can also click on 'Web' at the top of the page and find additional sites that might give you more info. You could also add + UK to the 'Web' version and get sites specific to the UK.
Here's two UK search engines that might be helpful to search for other annual plants, including annual vines. Don't hesitate to ask if you have any questions about any. There are so many annuals I felt it might be better for you to use a UK search engine as not all plants are available there and here in the US.
Let me know if you have more questions.