Comparison of those two solutions you mentioned above:
1. convert L to F without leaving.
As what you said, you may have all of the relative documents such as registration of temporary residence and photo, which will be charged almost 3700RMB for the conversion for 6 months.
(Not including the government fee)

2. apply F visa in UK
First of all, you should apply a business invitation letter before you apply the F visa in Chinese Embassy.
you can go to HK as well. There is a Chinese Embassy that can offer visa application services.
The letter would be cost 600RMB. Counting in the tickets， and the extension when you come back, the cost would be equal to the conversion.

Anyway, you can make the comparison which may help you out to choose which solution is better.

Good Luck
Grace

Hi,

you canot change L to F here in China by urself but an agency's help, but if u have F invitation letter, u can get F visa outside China mainland, HK or foreign countries r okay. However, normally with invitation letter, u will either get a short-period F visa n ul need to extend in China or ul get a long-term F with limitation on each stay. But even its like that, F extension is cheaper than L-F from agency.

Need my help or more information, Il b glad to help.

Phyllis