Is it possible to invest in the LTSE itself?
Wefunder or Republic or another such service would create a special purpose LLC to be the shareholder of record (or SAFE/noteholder or whatever) on your cap table.
What you're describing however is the way AngelList works.
The reason for this is mainly because the Reg CF terms prohibit crowdfunding investment vehicles (such as funds or SPVs).
(Source: I am former CSO of Republic)
Recently though I bought shares in a private company through wefunder and the docs specify I'll be a unitholder in an SPV LLC set up by wefunder that will own the stock, so I wonder if this prohibition changed with the $5m cap change?