I'm not at all sure that you recognize the process here. Generally the bacteria are used for amplification of the plasmid to levels at which you would use it. Doing this by PCR is technically possible, but much much more expensive, has a higher error rate (due to error prone DNA polymerases), and requires extra steps. You also won't get as much DNA out as you would from a bacterial culture. Culture is also scalable - you want more DNA, just grow more bacteria. Once you have a plasmid transformed into a bacterial strain, you can store stocks of this at -80 in glycerol form. You can then scrape a small bit, grow overnight and perform a <2h procedure the next day and end up with several micrograms of very clean DNA.
I highly recommend you read a good cloning protocol, such as those found in Sambrook et al., Molecular Cloning: a laboratory manual or Current protocols. Cloning has been around for >50 years now, I think the process is more or less optimized.
The usual process of generating a plasmid goes like this:
- Generate insert (often by PCR, but maybe from another plasmid or via a library generation)
- Digest insert with appropriate restriction enzymes
- Digest backbone with the same enzymes
- Ligate backbone and insert
- Transform into bacteria
- Pick colonies
- Screen colonies for insert (See below)
- Grow up colonies with successful clones
- Prepare plasmid DNA from successful clones
- Use that plasmid in experiments.
The bacteria will generally digest the linear DNA through production of exogenous DNases, and linear DNA often doesn't transform well, so your exo-nuclease step is unnecessary.
Usually when you are screening colonies you use primers specific for the insert, or, if wanting to determine if the insert has been inserted in a specific orientation, then one primer on the backbone and another on the insert. Only amplification from both primers producing a band of the correct size is a positive insert. In this case it is possible to produce primers that only recognize those plasmids that have an insert.
There are a few cases where you might generate primers that span the whole insert - sequencing of the whole insert might be one. Plasmids often have these sorts of sites already incorporated. If you see something labeled M13 or BGH, these are common sites for sequencing primers. Another common usage is if you want a range of inserts (e.g library preparation), and you don't know how big the inserts might be, then you can use the insert spanning primers to identify plasmids which have self-ligated (no insert) as these will only produce very small products on a gel. It is not possible to make insert spanning primers that only recognize the recombinants. No need for ligation or exonucleases.
Another possible scenario is where you want to change a base (or more than 1) in an already constructed plasmid. This is called Site Directed Mutagenesis. In this case you do want the PCR to go all the way around the plasmid, and parental DNA (from the plasmid) is damaged by digestion with the methylation specific restriction enzyme DpnI (PCR products are not methylated, but bacterially grown usually are), and the nicked circular product repaired by the bacteria after transformation.
Primers will not affect the GC content of your product. If you don't like the GC content of your primers, move them to a more amenable location.
It used to be that everyone did an alkaline phosphatase (APs) treatment of the digested DNA to prevent self-ligation. In practice, APs are very hard to get rid of or inactivate and tend to remove the necessary phosphate from your insert when you mix the products together at ligation. This leads to inhibition of the ligation and lowers your chance of success.