I don't have any direct experience of of LGPS but as I see it, it's not dissimilar to any other defined benefit scheme - most of which have closed their doors to new entrants (certainly in the private sector) - they're worth their weight in gold!!.
Here's my own experience of an AVC - when I left the US 25+ years ago, I cashed in my 401(K), paid the early withdrawal penalty (I recall 10%) and then invested the proceeds directly in a UK AVC (with Equitable Life - another story) - at the time I was in a DB scheme so the AVC seemed a sensible option for me to invest a modest lump sum and get 40% UK tax relief on the contribution.
Roll on 25 Years, the AVC had grown (there was no employer match and no further contributions from me) and I chose to split the proceeds, taking the maximum tax free amount and buying additional DB pension (effectively an annuity) with the remainder - not wanting to pay 40% tax otherwise. The value of the AVC was not huge (less than £25k, but a good value in comparison to the £4k I originally put in) but what I found surprising was that the UK tax free element was significantly more than the usual 25% - quite a complex area because of lifetime maximums and pre-A day rules.
See
https://www.gov.uk/tax-on-your-private-pension/lifetime-allowance for more information on lifetime allowances.
In terms of US tax of the lump sum, I declared 100% as taxable income because even then I was covered by FTC's and although it maybe I could have declared less, I deemed it not worth the hassle of a potential disagreement with the IRS - one of those is enough to last a lifetime, even if I did win my particular audit!
What I would say to the OP is get UK advice - Government DB schemes can be and have been lucrative - but there are pitfalls also if you breach the lifetime limits in terms of value. One of the reasons that so many NHS doctors have chosen to retire early is they've found that their pensions have breached lifetime allowances and UK income tax payable was quite penal (55% tax).
The other consideration is the annual allowance which is capped at £40k - but for DB schemes, this isn't the amount of your & your employer contributions but the increase in value of the scheme - a complex area.
The Pensions Advisory Service is a good place to start to get a more in depth understanding.