I believe the Orthrus brothers need to return

PAINTER'S TAPE to Hercules's mother.

As I noted in the comments,

for each word count from 4 to 9 inclusive, there are exactly two meats with that word count. This allows us to pair up the countries identified by the flags like so:

 4: SPAIN (PORK) 
    PANAMA (BEEF) 
 
 5: SLOVENIA (BACON) 
    CHINA (STEAK) 
 
 6: MONTENEGRO (TURKEY) 
    ETHIOPIA (SALMON) 
 
 7: ISRAEL (CHICKEN) 
    SRI LANKA (VENISON) 
 
 8: COSTA RICA (WILD BOAR) 
    CROATIA (SAUSAGES) 
 
 9: NEPAL (BRATWURST) 
    PERU (PARTRIDGE) 

The only information we haven't used yet is

the arrows in each country + meat box. If we interpret a right-pointing arrow to mean "forward" and a left-pointing arrow to mean "backward," we can modify our country strings based on which arrow is coming off of it (for example, NEPAL would be LAPEN since it has a left-pointing arrow coming off of it):

 SPAIN 
 PANAMA 
 
 AINEVOLS 
 CHINA 
 
 MONTENEGRO 
 AIPOIHTE 
 
 LEARSI 
 AKNAL IRS 
 
 COSTA RICA 
 AITAORC 
 
 LAPEN 
 PERU 

Finally, note that

between each pair of modified strings, there is a shared bigram. Taking that bigram and ordering based on meat word count length gives PAINTERS TAPE, which is appropriate because the brothers were decorating their apartment presumably using paint and would need to use painter's tape to ensure that the paint jobs are clean.

 SPAIN 
 PANAMA 
 
 AINEVOLS 
 CHINA 
 
 MONTENEGRO 
 AIPOIHTE 
 
 LEARSI 
 AKNAL IRS 
 
 COSTA RICA 
 AITAORC 
 
 LAPEN 
 PERU 

I believe the Orthrus brothers need to return

PAINTER'S TAPE to Hercules's mother.

As I noted in the comments,

for each word count from 4 to 9 inclusive, there are exactly two meats with that word count. This allows us to pair up the countries identified by the flags like so:

 4: SPAIN (PORK) 
    PANAMA (BEEF) 
 
 5: SLOVENIA (BACON) 
    CHINA (STEAK) 
 
 6: MONTENEGRO (TURKEY) 
    ETHIOPIA (SALMON) 
 
 7: ISRAEL (CHICKEN) 
    SRI LANKA (VENISON) 
 
 8: COSTA RICA (WILD BOAR) 
    CROATIA (SAUSAGES) 
 
 9: NEPAL (BRATWURST) 
    PERU (PARTRIDGE) 

The only information we haven't used yet is

the arrows in each box. If we interpret a right-pointing arrow to mean "forward" and a left-pointing arrow to mean "backward," then we can modify our country strings based on which arrow is coming off of the respective flag (for example, NEPAL would become LAPEN since its flag has a left-pointing arrow coming off of it):

 SPAIN 
 PANAMA 
 
 AINEVOLS 
 CHINA 
 
 MONTENEGRO 
 AIPOIHTE 
 
 LEARSI 
 AKNAL IRS 
 
 COSTA RICA 
 AITAORC 
 
 LAPEN 
 PERU 

Finally, note that

between each pair of modified strings, there is a shared bigram. Taking that bigram and ordering based on meat word count gives PAINTERS TAPE, which is appropriate because the brothers were decorating their apartment - presumably using paint - and would've needed to use painter's tape to ensure that the paint jobs were clean.

 SPAIN 
 PANAMA 
 
 AINEVOLS 
 CHINA 
 
 MONTENEGRO 
 AIPOIHTE 
 
 LEARSI 
 AKNAL IRS 
 
 COSTA RICA 
 AITAORC 
 
 LAPEN 
 PERU 

I believe the Orthurus brothers need to return

PAINTER'S TAPE to Hercules's mother.

As I noted in the comments,

for each word count from 4 to 9 inclusive, there are exactly two meats with that word count. This allows us to pair up the countries identified by the flags like so:

 4: SPAIN (PORK) 
    PANAMA (BEEF) 
 
 5: SLOVENIA (BACON) 
    CHINA (STEAK) 
 
 6: MONTENEGRO (TURKEY) 
    ETHIOPIA (SALMON) 
 
 7: ISRAEL (CHICKEN) 
    SRI LANKA (VENISON) 
 
 8: COSTA RICA (WILD BOAR) 
    CROATIA (SAUSAGES) 
 
 9: NEPAL (BRATWURST) 
    PERU (PARTRIDGE) 

The only information we haven't used yet is

the arrows in each country + meat box. If we interpret a right-pointing arrow to mean "forward" and a left-pointing arrow to mean "backward," we can modify our country strings based on which arrow is coming off of it (for example, NEPAL would be LAPEN since it has a left-pointing arrow coming off of it):

 SPAIN 
 PANAMA 
 
 AINEVOLS 
 CHINA 
 
 MONTENEGRO 
 AIPOIHTE 
 
 LEARSI 
 AKNAL IRS 
 
 COSTA RICA 
 AITAORC 
 
 LAPEN 
 PERU 

Finally, note that

between each pair of modified strings, there is a shared bigram. Taking that bigram and ordering based on meat word count length gives PAINTERS TAPE, which is appropriate because the brothers were decorating their apartment presumably using paint and would need to use painter's tape to ensure that the paint jobs are clean.

 SPAIN 
 PANAMA 
 
 AINEVOLS 
 CHINA 
 
 MONTENEGRO 
 AIPOIHTE 
 
 LEARSI 
 AKNAL IRS 
 
 COSTA RICA 
 AITAORC 
 
 LAPEN 
 PERU 

I believe the Orthrus brothers need to return

PAINTER'S TAPE to Hercules's mother.

As I noted in the comments,

for each word count from 4 to 9 inclusive, there are exactly two meats with that word count. This allows us to pair up the countries identified by the flags like so:

 4: SPAIN (PORK) 
    PANAMA (BEEF) 
 
 5: SLOVENIA (BACON) 
    CHINA (STEAK) 
 
 6: MONTENEGRO (TURKEY) 
    ETHIOPIA (SALMON) 
 
 7: ISRAEL (CHICKEN) 
    SRI LANKA (VENISON) 
 
 8: COSTA RICA (WILD BOAR) 
    CROATIA (SAUSAGES) 
 
 9: NEPAL (BRATWURST) 
    PERU (PARTRIDGE) 

The only information we haven't used yet is

the arrows in each country + meat box. If we interpret a right-pointing arrow to mean "forward" and a left-pointing arrow to mean "backward," we can modify our country strings based on which arrow is coming off of it (for example, NEPAL would be LAPEN since it has a left-pointing arrow coming off of it):

 SPAIN 
 PANAMA 
 
 AINEVOLS 
 CHINA 
 
 MONTENEGRO 
 AIPOIHTE 
 
 LEARSI 
 AKNAL IRS 
 
 COSTA RICA 
 AITAORC 
 
 LAPEN 
 PERU 

Finally, note that

between each pair of modified strings, there is a shared bigram. Taking that bigram and ordering based on meat word count length gives PAINTERS TAPE, which is appropriate because the brothers were decorating their apartment presumably using paint and would need to use painter's tape to ensure that the paint jobs are clean.

 SPAIN 
 PANAMA 
 
 AINEVOLS 
 CHINA 
 
 MONTENEGRO 
 AIPOIHTE 
 
 LEARSI 
 AKNAL IRS 
 
 COSTA RICA 
 AITAORC 
 
 LAPEN 
 PERU