Is my restore process being badly affected by this third-party non copy-only backup?
Maybe this will help you understand. Let's do a demo! We'll make a dummy database with some dummy data.

USE master;

/*Create a dummy database*/
CREATE DATABASE LogRestoreTest

/*We full now*/
ALTER DATABASE LogRestoreTest SET RECOVERY FULL

/*Context is everything*/
USE LogRestoreTest

/*If nothing changes, do we even need a log backup?*/
CREATE TABLE dbo.t1 (Id INT)


Now we'll take one full backup. Just one. Promise.

/*Take a full backup, dummy*/

BACKUP DATABASE LogRestoreTest 
TO DISK = 'F:\Backup\LRT_FULL.bak' 
WITH INIT, FORMAT, COMPRESSION


Now we'll make some changes and take some log backups. Just like real life. It's fun. No drinking yet.

/*Make a change*/
INSERT dbo.t1 (Id )
VALUES ( 1 )

/*Take a log backup*/
BACKUP LOG LogRestoreTest 
TO DISK = 'F:\Backup\LRT_LOG_1.trn' 
WITH INIT, FORMAT, COMPRESSION

/*Make another change*/
INSERT dbo.t1 (Id )
VALUES ( 2 )

/*Take another log backup*/
BACKUP LOG LogRestoreTest 
TO DISK = 'F:\Backup\LRT_LOG_2.trn' 
WITH INIT, FORMAT, COMPRESSION

/*Make another change*/
INSERT dbo.t1 (Id )
VALUES ( 3 )


Now we'll take an 'out of band' full backup, and another log backup.

/*A second full backup appears!*/
BACKUP DATABASE LogRestoreTest 
TO DISK = 'F:\Backup\LRT_FULL_2.bak' 
WITH INIT, FORMAT, COMPRESSION

/*Take another log backup*/
BACKUP LOG LogRestoreTest 
TO DISK = 'F:\Backup\LRT_LOG_3.trn' 
WITH INIT, FORMAT, COMPRESSION


If I want to restore data to the third log backup, I have two options.

Restore the first full backup, and all three log backups:

/*Restore the full backup*/
RESTORE DATABASE LogRestoreTest
FROM DISK = 'F:\Backup\LRT_FULL.bak' 
WITH REPLACE, NORECOVERY


/*Square one*/
RESTORE DATABASE LogRestoreTest
FROM DISK = 'F:\Backup\LRT_LOG_1.trn' 
WITH NORECOVERY

/*What about to here?*/
RESTORE DATABASE LogRestoreTest
FROM DISK = 'F:\Backup\LRT_LOG_2.trn' 
WITH NORECOVERY

/*Obligatory Your Mom*/
RESTORE DATABASE LogRestoreTest
FROM DISK = 'F:\Backup\LRT_LOG_3.trn' 
WITH NORECOVERY


Or I can restore the most recent full and then the final log file:

/*Restore the full backup*/
RESTORE DATABASE LogRestoreTest
FROM DISK = 'F:\Backup\LRT_FULL_2.bak' 
WITH REPLACE, NORECOVERY


/*What happens if I try to jump the restores?*/
RESTORE DATABASE LogRestoreTest
FROM DISK = 'F:\Backup\LRT_LOG_3.trn' 
WITH NORECOVERY


You'll have to forgive me not explaining differential backups here, but there are plenty of resources out there that cover DBA 101 material.

Hope this helps!