What to cover the ground with between raised veg beds

I've put in some raised veg beds using sleepers over an existing lawn. I've now realised it's almost impossible to cut the grass between the beds though, as although the lawn mower fits the gaps it doesn't get close enough and I have long grass all around the edges.
I've thought about putting down weed control fabric and covering with bark chippings but I'm concerned about how the birds will fling that around and it end up everywhere. I've also thought about using the weed control fabric and gravel, but hubby feels that will turn into the local cat toilet.
Any body got any other ideas? I don't want to slab/pave it, not just because of the expense but also because it's a very boggy/clay garden already and i don't want to add to the drainage issues.
Do you think something like a creeping thyme would work? or maybe some other low growing plant that I could still walk on?
Thanks
I've thought about putting down weed control fabric and covering with bark chippings but I'm concerned about how the birds will fling that around and it end up everywhere. I've also thought about using the weed control fabric and gravel, but hubby feels that will turn into the local cat toilet.
Any body got any other ideas? I don't want to slab/pave it, not just because of the expense but also because it's a very boggy/clay garden already and i don't want to add to the drainage issues.
Do you think something like a creeping thyme would work? or maybe some other low growing plant that I could still walk on?
Thanks
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Eventually the chips break down into soil, and it acts as a great substrate for weeds, so more and more weeding is required, or replacing the chips every other year. We still think this is the best option for us, plus we get free bark chips at our allotments.
Personally I wouldn't use paving or anything hard, as it's a killer on the knees! Better something soft under-knee. Pea gravel would work, and be more comfortable than larger gravel/pebbles, but depending on how much you need, it could be pricey. In theory creeping Thyme might work, but if you are on heavy clay, there may not be enough drainage (agree with JennyJ that you would need to buy a lot of plants, or wait ages for it to cover everything).
I don't have a cat problem as the nearest houses are a third of a mile away, just the occasional farm cat is seen but has never used the gravel as a toilet. It's too difficult to scrape aside, and cats don't have gloves!
The slate would also work well.
Your budget will be a factor
I live in west central Scotland - not where that photo is...
@WhereAreMySecateurs even with shears it’s a really slow job trying to get close enough to the wood. The grass strands grow vertical up the side of the wood and you literally need to move each strand and cut with scissors to make it look anything like neat. I really should have thought about it a little more before I built the beds.
"We don't stop playing because we grow old; we grow old because we stop playing." - George Bernard Shaw