Assign a variable to a string, using only a string containing that variable's name

I have many .rda file, which themselves are correctly named, however the variables they each contain (only one per file) are incorrectly labelled. I am attempting to load them into the workspace one by one, rename them to the correct variable name and then save them into a new folder, using the same file name that each file already has.

My attempt is below and the problem is that in the step using assign, it seems to only be assign the actual name of the variable I'd like to be reassigning. So when I then save this and load it again, I have a variable holding a string, not the actual results.

input_folder <- "/path/to/input_folder/"
output_folder <- "/path/to/output_folder/"

## What I know
A <- "data_chunk"                     #input name
B <- "results_Lag.1"                  #variable/data name in workspace
C <- "results.L1.trad_small.rda"  #file saved name

load(paste0(input_folder, C))            #load the data into the workspace

to_save <- substr(C, 1, nchar(C)-4)     #desired new name in workspace. the -4 loses ".rda"

## Four variations on how to re-assign the data
assign(to_save, B)
assign(to_save, get(B))
assign(to_save, as.name(B))
assign(to_save, eval(parse(B)))

## Save the data with new vairable name to new folder
save(list = to_save, file = paste0(output_folder, C))

As you can see from the three variations, I have seen several ways ([1][2][3]) to call a variable from the workspace using a string.

The first three simply save my my string B as the variable, while the last variations using eval(parse()) returns an error:

> assign(to_save, eval(parse(B)))
Error in file(filename, "r") : cannot open the connection
Calls: assign -> eval -> parse -> file
In addition: Warning message:
In file(filename, "r") :
  cannot open file 'glm_results_Lag.1': No such file or directory
Calls: assign -> eval -> parse -> file

If I call get(B) in the console, it does return the contents of the variable as expected.

EDIT

As @joran says in the comments, the second variant does produce the expected result, saving the data to the specified string:

assign(to_save, get(B))

The size of the new object on disk, however, is different to that of the loaded file (As i mention in the comments).

I have no explanation for this...